#6444989

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	#24270	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)(15,20)(16,27)(17,22)(18,32)(19,23)(21,29)(24,38)(25,34)(26,35)(28,37)(30,41)(31,36)(33,39)(40,43)(42,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,24)(17,31)(18,25)(19,26)(21,33)(22,36)(23,35)(27,38)(28,42)(29,39)(30,40)(32,34)(37,44)(41,43);(2,4)(13,16,17,28,19,30,18,29)(14,21,32,41,23,37,22,27)(15,24,31,42,26,40,25,39)(20,33,34,43,35,44,36,38);(2,4)(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,28,30,29)(20,34,35,36)(21,27,37,41)(24,39,40,42)(33,43,44,38);(5,7,8,11)(6,10,9,12)(15,26)(20,35)(24,40)(25,31)(33,44)(34,36)(38,43)(39,42)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=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, 6352185);