#425181

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1056 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)(13,14,18,24)(15,26,29,21)(16,22,17,32)(19,25,20,36)(23,31,41,34)(27,42,43,37)(28,44,33,40)(30,39,35,38);(1,3,4,2)(5,7,9,12)(6,10,8,11)(13,15,18,29)(14,21,24,26)(16,28,17,33)(19,35,20,30)(22,40,32,44)(23,42,41,37)(25,39,36,38)(27,34,43,31);(2,3)(5,8,9,6)(7,11,12,10)(13,16,18,17)(14,22,24,32)(15,27,29,43)(19,34,20,31)(21,37,26,42)(23,36,41,25)(28,30,33,35)(38,44,39,40);(2,3)(7,12)(10,11)(13,17,18,16)(14,23,24,41)(15,28,29,33)(19,34,20,31)(21,37,26,42)(22,36,32,25)(27,30,43,35)(38,40,39,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ef,d^2=i,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, 332377);