C₂×#873858 #72299282

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

Aliases

direct product

C₂×#873858

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

30757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297586, C₂×#72297594, C₂×#72297560
maximal normal subgroup of	C₂×#72297586, C₂×#72297594, C₂×#72297560

GAP

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

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

C2×#873858 #72299282