#424638

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1240 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:

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

Polycyclic presentation:

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

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, 331834);