#6443488

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350684);