#147138

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54334);