#129595

order	512
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1790
commutator	C₂×Q₈	C₂²×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8316
prime radical	#3529	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

599 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,14,8,16,31,48,9,15,32,47,6,13,29,44)(3,17,20,50,21,34,55,59,22,41,30,57,19,49,53,61)(4,12,24,42,23,40,56,45,5,25,28,46,27,11,38,39)(10,33,36,51,35,58,60,26,37,54,62,52,43,63,64,18);(1,3)(2,10)(4,18)(5,26)(6,19)(7,30)(8,21)(9,22)(11,34)(12,41)(13,43)(14,36)(15,37)(16,35)(17,25)(20,32)(23,51)(24,54)(27,52)(28,33)(29,55)(31,53)(38,58)(39,59)(40,49)(42,57)(44,64)(45,61)(46,50)(47,62)(48,60)(56,63);(1,4)(2,11)(3,18)(5,9)(6,27)(7,28)(8,23)(10,34)(12,16)(13,25)(14,45)(15,40)(17,43)(19,52)(20,54)(21,51)(22,26)(24,32)(29,56)(30,33)(31,38)(35,41)(36,61)(37,49)(39,47)(42,44)(46,48)(50,60)(53,58)(55,63)(57,64)(59,62)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=eg,f^b=fgi,d^c=dgi,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239956, #72250047, #72250052, …
maximal normal subgroup of	#72239956, #72250047, #72250052, …
maximal quotient of	#72239961, #72279609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36791);