#146674

order	512
nilpotency class	4
isoclinism	isoclinic class of #12711
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10761
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

985 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 188 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317336, #72317395, #72317406
maximal normal subgroup of	#72317336, #72317395, #72317406

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53870);