C₂×#23355 #6407623

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#23355

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂⁴	C₂³×C₄
prime radical	C₂²×#127	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2934 conjugacy classes of subgroups; only the 1098 normal subgroups are shown.

normal subgroups only

This group has 2934 conjugacy classes of subgroups — too many to draw. Only its 1098 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);(3,4)(5,9)(6,10)(7,8)(11,12)(13,17)(14,16)(15,18)(19,20,21,23,22,24,25,26);(4,8)(5,6)(13,18)(15,17);(3,5,7,6)(4,9,8,10)(11,13,14,18)(12,15,16,17);(3,6,7,5)(4,9,8,10)(12,16)(15,17)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#6407637, #72264742
maximal normal subgroup of	C₂×#6407637, #72264742

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314819);