#358316

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265512);