C₂×#516752 #72325608

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#516752

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

55101 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232088, C₂×#72232094, C₂×#72232186
maximal normal subgroup of	C₂×#72232088, C₂×#72232094, C₂×#72232186

GAP

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

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

C2×#516752 #72325608