#128699

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

326 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)(9,11)(10,12)(13,14,17,21,18,15,22,24,16,19,25,27,26,23,20,28);(2,4)(5,7,9,12,10,11,8,6)(14,19)(15,23)(21,27)(24,28);(5,8,10,9)(6,7,12,11)(13,15,17,24,18,19,22,27,16,23,25,28,26,14,20,21)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fg,c^b=cgi

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, 35895);