#6445110

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	#25109	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2069 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)(7,10)(8,9)(11,12)(13,14,17,19)(15,23,26,38)(16,21,29,33)(18,31,24,22)(20,34,32,27)(25,39,28,42)(30,44,40,37)(35,43,36,41);(1,2)(3,4)(6,9)(10,12)(14,19)(20,32)(21,33)(22,31)(23,38)(35,36)(37,44)(39,42);(2,4)(13,15,24,34,17,26,18,27)(14,20,31,38,19,32,22,23)(16,28,40,41,29,25,30,43)(21,36,44,39,33,35,37,42);(2,4)(13,16)(14,21)(15,25)(17,29)(18,30)(19,33)(20,35)(22,37)(23,39)(24,40)(26,28)(27,41)(31,44)(32,36)(34,43)(38,42);(5,7,8,11)(6,10,9,12)(15,26)(16,29)(20,32)(21,33)(23,38)(27,34)(30,40)(37,44)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6352306);