#7156036

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 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)(15,24)(16,22)(17,23)(18,20)(19,33)(21,28)(25,41)(26,34)(27,40)(29,35)(30,44)(31,38)(32,39)(36,42)(37,43);(2,3)(6,9)(7,11)(14,20)(16,29)(21,34)(23,33)(24,40)(25,30)(31,32)(42,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,28,18,27,17,26)(14,21,23,24,20,34,33,40)(16,30,31,43,29,25,32,42)(22,37,39,44,35,36,38,41);(1,3,4,2)(6,9)(10,12)(13,16,18,29)(14,22,20,35)(15,25,27,30)(17,32,19,31)(21,36,34,37)(23,39,33,38)(24,41,40,44)(26,43,28,42);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,20,33)(15,26,27,28)(16,31,29,32)(21,24,34,40)(22,38,35,39)(25,42,30,43)(36,44,37,41)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,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, 7063232);