#6685647

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	#42673	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1908 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6592843);