C₂×#536623 #72325147

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

Aliases

direct product

C₂×#536623

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

48969 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232094, C₂×#72232084, C₂×#72232121, …
maximal normal subgroup of	C₂×#72232094, C₂×#72232084, C₂×#72232121, …

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

C2×#536623 #72325147