#7156016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(7,11)(10,12)(13,14)(15,23)(16,21)(17,19)(18,30)(20,27)(22,24)(25,36)(26,35)(28,32)(29,41)(31,37)(33,40)(34,38)(39,43)(42,44);(2,3)(6,9)(7,11)(14,19)(16,28)(20,31)(22,30)(23,35)(25,39)(29,38)(40,44);(5,6)(7,10)(8,9)(11,12)(13,15,24,37,17,26,18,27)(14,20,30,35,19,31,22,23)(16,25,38,44,28,39,29,40)(21,33,41,43,32,42,34,36);(1,3,4,2)(6,9)(10,12)(13,16,17,28)(14,21,19,32)(15,25,26,39)(18,29,24,38)(20,33,31,42)(22,34,30,41)(23,36,35,43)(27,40,37,44);(5,7,8,11)(6,10,9,12)(16,28)(21,32)(25,39)(29,38)(33,42)(34,41)(36,43)(40,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063212);