#360044

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,4)(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,24)(21,23)(22,28)(25,30)(27,32)(29,31);(10,13)(14,15)(17,19)(18,22)(20,28)(21,27)(23,31)(24,26)(25,32)(29,30);(2,4)(18,23)(19,27)(21,30)(24,25)(28,29)(31,32);(9,11,12,16)(10,14,13,15)(18,24)(22,26)(23,25)(31,32)

Polycyclic presentation:

2^9|a^2=h,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308250, #72308259, #72308297
maximal normal subgroup of	#72308250, #72308259, #72308297

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267240);