#7054178

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	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2113 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)(15,24)(16,21)(17,22)(18,32)(19,23)(20,26)(25,41)(27,35)(28,44)(29,38)(30,40)(31,39)(33,36)(34,43)(37,42);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12);(5,7,8,12)(6,9,10,11)(13,15,19,27)(14,20,23,36)(16,28,30,25)(17,26,18,33)(21,37,40,34)(22,35,32,24)(29,43,31,42)(38,44,39,41);(2,4)(7,12)(9,11)(13,16,19,30)(14,21,23,40)(15,25,27,28)(17,31,18,29)(20,34,36,37)(22,39,32,38)(24,41,35,44)(26,43,33,42);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,26,27,33)(16,29,30,31)(20,35,36,24)(21,38,40,39)(25,42,28,43)(34,41,37,44)

Polycyclic presentation:

2^9|c^2=hi,d^2=i,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, 6961374);