#416534

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

1176 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=fh,c^2=e,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,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, 323730);