C₂×#524781 #72324434

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#524781

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

66069 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232044, C₂×#72232053, C₂×#72232057, …
maximal normal subgroup of	C₂×#72232044, C₂×#72232053, C₂×#72232057, …

GAP

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

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

C2×#524781 #72324434