C₂×#617710 #72295211

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂×#617710

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

107597 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72237456, #72323764, #72323765, …
maximal normal subgroup of	C₂×#72237456, #72323764, #72323765, …
maximal quotient of	C₂×#72293963, C₂×#72332554, C₂×#72298486, …

GAP

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

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

C2×#617710 #72295211