C₂×#2760379 #72325262

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

Aliases

direct product

C₂×#2760379

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

41669 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232087, C₂×#72232147, C₂×#72232105
maximal normal subgroup of	C₂×#72232087, C₂×#72232147, C₂×#72232105

GAP

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

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

C2×#2760379 #72325262