#391938

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,19,20,18)(21,22,25,26)(23,27,24,28);(1,2)(3,4)(6,9)(7,11)(14,17)(18,19)(21,23,26,28,25,24,22,27);(2,4)(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,23)(22,27,26,28);(5,7,8,11)(6,10,9,12)(15,20)(18,19)(21,22,25,26)(23,28,24,27)

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299134);