C₂×#519127 #72331095

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#519127

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

70593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72264092, C₂×#72323286, C₂×#72261595
maximal normal subgroup of	C₂×#72264092, C₂×#72323286, C₂×#72261595

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((12,14), (8,10)(11,12)(13,14)(18,22)(20,21), (8,10)(15,16,19,17)(18,21,22,20), (1,2), (3,4,5,6)(7,8,9,10)(15,17,19,16)(18,20,22,21), (15,18)(16,20)(17,21)(19,22));

C2×#519127 #72331095