C₂×#8308 #184759

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

Aliases

direct product

C₂×#8308

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

4146 conjugacy classes of subgroups; only the 434 normal subgroups are shown.

normal subgroups only

This group has 4146 conjugacy classes of subgroups — too many to draw. Only its 434 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) = 30
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 30 points:

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

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, 91955);