C₂×#674382 #72325317

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#674382

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

59041 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)(8,9)(12,13)(15,16,19,21)(17,20,22,18);(4,5)(7,8)(9,10)(12,13)(15,17,19,22)(16,20,21,18);(17,22)(18,20);(4,5)(8,9)(11,12,14,13)(15,18,19,20)(16,17,21,22);(1,2);(3,5,6,4)(7,9,10,8)(11,13,14,12)(15,17)(16,18)(19,22)(20,21)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232096, C₂×#72232108, C₂×#72232163, …
maximal normal subgroup of	C₂×#72232096, C₂×#72232108, C₂×#72232163, …

GAP

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

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

C2×#674382 #72325317