C₂×#837030 #72302527

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

Aliases

direct product

C₂×#837030

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2397	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

29027 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280318, C₂×#72279903, C₂×#72279775, …
maximal normal subgroup of	C₂×#72280318, C₂×#72279903, C₂×#72279775, …

GAP

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

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

C2×#837030 #72302527