C₂×#726574 #72325622

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

Aliases

direct product

C₂×#726574

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

42433 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)(11,12,15,13)(14,17,18,16)(19,20)(21,25)(22,23)(24,26);(4,6)(8,10)(12,13)(16,17)(20,23)(24,25);(12,13)(14,18)(20,23)(21,26);(1,2);(4,6)(7,8)(9,10)(11,13,15,12)(14,16,18,17)(19,21,22,26)(20,24,23,25);(3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,20)(21,25)(22,23)(24,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232116, C₂×#72232131, C₂×#72232170
maximal normal subgroup of	C₂×#72232116, C₂×#72232131, C₂×#72232170

GAP

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

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

C2×#726574 #72325622