C₂×#772944 #72299425

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#772944

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

27175 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72282047, C₂×#72297560
maximal normal subgroup of	C₂×#72282047, C₂×#72297560

GAP

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

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

C2×#772944 #72299425