C₂×#154248 #72334583

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

Aliases

direct product

C₂×#154248

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂⁴	C₂⁴×C₄
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

81997 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

(3,4,5,6)(7,8,9,10)(12,14)(16,18)(20,22);(4,6)(7,8,9,10)(12,14)(19,20)(21,22);(11,12)(13,14)(15,16,17,18);(1,2);(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×#72332555, C₂×#72298481, C₂×#72334303, …
maximal normal subgroup of	C₂×#72332555, C₂×#72298481, C₂×#72334303, …
maximal quotient of	#72335108

GAP

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

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

C2×#154248 #72334583