#7054133

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961329);