C₂×#1515009 #72324977

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#1515009

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

41493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(11,12)(13,15)(14,16)(17,18)(19,20)(21,24)(22,23)(25,26);(3,4)(5,6)(8,10)(13,17)(15,18)(20,23)(24,25);(4,6)(11,12,14,16)(13,15,17,18)(19,20,22,23)(21,24,26,25);(7,8,9,10)(11,12)(13,18)(14,16)(15,17)(19,21)(20,24)(22,26)(23,25);(1,2);(3,4)(5,6)(11,13)(12,15)(14,17)(16,18)(19,20)(21,25)(22,23)(24,26)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232168, C₂×#72232123, C₂×#72232103, …
maximal normal subgroup of	C₂×#72232168, C₂×#72232123, C₂×#72232103, …

GAP

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

gap> G := Group((11,12)(13,15)(14,16)(17,18)(19,20)(21,24)(22,23)(25,26), (3,4)(5,6)(8,10)(13,17)(15,18)(20,23)(24,25), (4,6)(11,12,14,16)(13,15,17,18)(19,20,22,23)(21,24,26,25), (7,8,9,10)(11,12)(13,18)(14,16)(15,17)(19,21)(20,24)(22,26)(23,25), (1,2), (3,4)(5,6)(11,13)(12,15)(14,17)(16,18)(19,20)(21,25)(22,23)(24,26));

C2×#1515009 #72324977