C₂×#637061 #72299391

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#637061

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

33393 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297599, C₂×#72297594, C₂×#72297397
maximal normal subgroup of	C₂×#72297599, C₂×#72297594, C₂×#72297397

GAP

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

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

C2×#637061 #72299391