C₂×#10392 #238071

order	512
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#10392

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2342 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 2342 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#251859, C₂×#251828, #72284473, …
maximal normal subgroup of	C₂×#251859, C₂×#251828, #72284473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145267);