#6467173

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6374369);