#6682605

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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	#37659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2263 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)(7,11)(8,10)(9,12)(13,14,19,23)(15,20,26,36)(16,27,29,37)(17,22,18,32)(21,31,38,28)(24,39,41,44)(25,34,30,35)(33,40,43,42);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,30)(18,25)(19,26)(21,33)(22,35)(23,36)(27,39)(28,42)(29,41)(31,40)(32,34)(37,44)(38,43);(2,4)(13,16,18,28,19,29,17,31)(14,21,22,27,23,38,32,37)(15,24,25,42,26,41,30,40)(20,33,35,39,36,43,34,44);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,25,26,30)(16,28,29,31)(20,34,36,35)(21,37,38,27)(24,40,41,42)(33,39,43,44);(2,4)(7,12)(9,11)(16,29)(21,38)(24,41)(27,37)(28,31)(33,43)(39,44)(40,42)

Polycyclic presentation:

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

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, 6589801);