#6491383

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3516	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398579);