#6685189

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1872 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592385);