#128282

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄×C₈	C₂³
commutator	C₂²	C₂×C₄×C₁₆
socle	C₂³	C₂×C₄×C₈
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ch

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35478);