C₂×#839418 #72302551

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

Aliases

direct product

C₂×#839418

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

31399 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)(3,4)(11,12,16,14)(13,15,18,17)(20,22)(24,25);(7,8,9,10)(11,12)(13,15)(14,16)(17,18)(19,20,23,22)(21,24,26,25);(3,5,4,6)(11,13,16,18)(12,17,14,15)(20,22)(24,25);(5,6)(7,8)(9,10)(12,14)(15,17)(19,20,23,22)(21,25,26,24);(11,14,16,12)(13,15,18,17)(19,21,23,26)(20,24,22,25);(3,6,4,5)(7,8)(9,10)(11,15)(12,18)(13,14)(16,17)(19,22,23,20)(21,24,26,25)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280318, C₂×#72279893, C₂×#72279775, …
maximal normal subgroup of	C₂×#72280318, C₂×#72279893, C₂×#72279775, …

GAP

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

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

C2×#839418 #72302551