#419414

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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1378 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ei,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,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, 326610);