#426619

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=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, 333815);