#398484

order	512
nilpotency class	3
isoclinism	isoclinic class of #394329
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305680);