C₂×#154194 #72289558

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#154194

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂⁴	C₂⁴×C₄
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

166661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72286421, C₂×#72286425, C₂×#72250275, …
maximal normal subgroup of	C₂×#72286421, C₂×#72286425, C₂×#72250275, …

GAP

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

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

C2×#154194 #72289558