#6385074

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=dfi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6292270);