C₂×#617361 #72334522

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#617361

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

78561 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72330134, C₂×#72330145, C₂×#72334295
maximal normal subgroup of	C₂×#72330134, C₂×#72330145, C₂×#72334295
maximal quotient of	C₂×#72279059

GAP

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

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

C2×#617361 #72334522