#427371

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1259 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)(7,10)(8,9)(11,12)(13,14,17,23)(15,25,28,20)(16,21,31,41)(18,24,19,35)(22,34,42,32)(26,37,43,36)(27,40,30,44)(29,39,33,38);(1,2)(3,4)(13,15,17,28)(14,20,23,25)(16,30,31,27)(18,33,19,29)(21,40,41,44)(22,36,42,37)(24,39,35,38)(26,34,43,32);(6,9)(10,12)(13,16)(14,21)(15,26)(17,31)(18,34)(19,32)(20,36)(22,35)(23,41)(24,42)(25,37)(27,29)(28,43)(30,33)(38,44)(39,40);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,27)(17,31)(18,32)(19,34)(20,37)(21,35)(23,42)(24,41)(25,36)(26,29)(28,30)(33,43)(38,44)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,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, 334567);