C₂×#8703 #189696

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#8703

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1474 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 1474 conjugacy classes of subgroups — too many to draw. Only its 338 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,14,18,15,17,13,16)(19,20)(21,24)(22,29,30,25)(23,26)(27,33,34,31)(28,32);(4,7)(8,10)(11,13,15,14)(12,16,17,18)(19,21,23,28)(20,24,26,32)(22,27,30,34)(25,31,29,33);(3,5,6,9)(4,8,7,10)(12,17)(13,14)(19,22)(20,25)(21,27)(23,30)(24,31)(26,29)(28,34)(32,33)

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, 96892);