#7054196

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^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=eh,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961392);