#381151

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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

1330 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fi,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, 288347);