#7156042

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#45387	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2056 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)(16,27)(17,22)(18,24)(19,20)(21,30)(25,37)(26,38)(28,43)(29,34)(31,35)(32,44)(33,40)(36,42)(39,41);(2,3)(6,9)(7,11)(13,15,18,19)(14,20,24,23)(16,25,29,30)(17,31,33,26)(21,34,37,27)(22,38,40,35)(28,42,32,41)(36,43,39,44);(5,6)(7,10)(8,9)(11,12)(13,16,15,25,18,29,19,30)(14,21,20,34,24,37,23,27)(17,32,26,42,33,28,31,41)(22,39,35,43,40,36,38,44);(1,3,4,2)(6,9)(10,12)(13,17,18,33)(14,22,24,40)(15,26,19,31)(16,28,29,32)(20,35,23,38)(21,36,37,39)(25,41,30,42)(27,43,34,44);(5,7,8,11)(6,10,9,12)(13,15,18,19)(14,23,24,20)(16,25,29,30)(17,31,33,26)(21,27,37,34)(22,35,40,38)(28,42,32,41)(36,44,39,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gi,d^2=f,e^2=hi,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=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063238);