#6440164

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#24438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2175 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)(13,14,18,23)(15,19,26,35)(16,21,27,37)(17,29,31,22)(20,30,36,28)(24,33,38,44)(25,40,42,34)(32,41,43,39);(1,2)(3,4)(6,9)(10,12)(13,15,18,26)(14,19,23,35)(16,24,27,38)(17,25,31,42)(20,32,36,43)(21,33,37,44)(22,34,29,40)(28,39,30,41);(2,4)(13,16,18,27)(14,20,23,36)(15,24,26,38)(17,30,31,28)(19,32,35,43)(21,22,37,29)(25,41,42,39)(33,34,44,40);(2,4)(13,16)(14,21)(15,24)(17,28)(18,27)(19,33)(20,29)(22,36)(23,37)(25,39)(26,38)(30,31)(32,40)(34,43)(35,44)(41,42);(5,7,8,11)(6,10,9,12)(16,27)(20,36)(21,37)(24,38)(28,30)(32,43)(33,44)(39,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347360);