#6514220

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1790 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421416);