C₂×#154671 #72267185

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#154671

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂⁴	C₂⁴×C₄
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁴×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

167573 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72210047, C₂×#72286418, C₂×#72252084, …
maximal normal subgroup of	C₂×#72210047, C₂×#72286418, C₂×#72252084, …
maximal quotient of	C₂×#72219227, #72337398

GAP

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

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

C2×#154671 #72267185