C₂×#21684 #6373707

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#21684

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂⁴	C₂³×C₄
prime radical	C₂×#2397	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3530 conjugacy classes of subgroups; only the 1058 normal subgroups are shown.

normal subgroups only

This group has 3530 conjugacy classes of subgroups — too many to draw. Only its 1058 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) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280903);