C₂×#8306 #184756

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#8306

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

2618 conjugacy classes of subgroups; only the 418 normal subgroups are shown.

normal subgroups only

This group has 2618 conjugacy classes of subgroups — too many to draw. Only its 418 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) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(3,4)(5,6)(7,8)(9,15,17,12)(10,13)(11,14)(16,21,22,19)(18,20)(23,24,26,29,27,30,33,36)(25,28,32,35,31,34,37,38);(7,9,11,17)(8,12,14,15)(10,16,18,22)(13,19,20,21)(23,25)(24,28)(26,32)(27,31)(29,35)(30,34)(33,37)(36,38);(4,6)(7,10,11,18)(8,13,14,20)(9,16,17,22)(12,19,15,21)(25,31)(28,34)(32,37)(35,38)

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, 91952);