C₂×#2921513 #72325495

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#2921513

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

53775 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)(12,14)(15,16)(17,21)(18,20)(19,22);(7,8)(9,10)(15,17)(16,19)(18,22)(20,21);(8,10)(11,12,13,14)(16,20)(17,22);(4,6)(12,14)(15,17)(16,21)(18,22)(19,20);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,22)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232074, C₂×#72232101, C₂×#72232181
maximal normal subgroup of	C₂×#72232074, C₂×#72232101, C₂×#72232181

GAP

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

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

C2×#2921513 #72325495