C₂×#918025 #72323868

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

Aliases

direct product

C₂×#918025

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

57157 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)(13,16)(14,15)(17,18);(3,4)(5,6)(8,10)(13,17)(16,18)(19,20,22,24)(21,25,26,23);(7,8)(9,10)(12,15)(16,18)(19,20,22,24)(21,23,26,25);(4,5)(11,12)(13,18)(14,15)(16,17)(19,20)(21,23)(22,24)(25,26);(1,2);(3,5,6,4)(11,13,14,17)(12,16,15,18)(19,21)(20,23)(22,26)(24,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323252, C₂×#72323250, C₂×#72323256, …
maximal normal subgroup of	C₂×#72323252, C₂×#72323250, C₂×#72323256, …

GAP

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

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

C2×#918025 #72323868