#146432

order	512
nilpotency class	4
isoclinism	isoclinic class of #126701
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1735
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10721
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

869 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 264 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)(3,15)(4,21,23,10)(5,26)(6,11)(7,31)(8,12)(9,17)(13,29)(14,28)(16,43,45,41)(18,35)(19,50)(20,39)(22,37,57,53)(24,52,58,36)(25,38,56,54)(27,33)(30,40)(32,44,51,55)(34,48)(42,49)(46,61,64,60)(47,59,63,62);(1,3,6,18,7,19,29,17,8,20,30,49,28,48,5,27)(2,9,13,34,14,35,11,39,12,33,26,50,31,42,40,15)(4,16,24,46,25,47,57,44,23,45,58,64,56,63,22,55)(10,32,37,59,38,60,36,43,21,51,53,62,54,61,52,41);(1,4)(2,10)(3,16)(5,22)(6,24)(7,25)(8,23)(9,32)(11,36)(12,21)(13,37)(14,38)(15,41)(17,44)(18,46)(19,47)(20,45)(26,53)(27,55)(28,56)(29,57)(30,58)(31,54)(33,51)(34,59)(35,60)(39,43)(40,52)(42,61)(48,63)(49,64)(50,62)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,h^a=hi,d^b=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307216, #72307247, #72307287, …
maximal normal subgroup of	#72307216, #72307247, #72307287, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53628);