C₂×#648270 #72337976

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

Aliases

direct product

C₂×#648270

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

47563 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232291, C₂×#72232295, C₂×#72232311
maximal normal subgroup of	C₂×#72232291, C₂×#72232295, C₂×#72232311

GAP

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

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

C2×#648270 #72337976