#413297

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 320493);