C₂×#851176 #72337964

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

Aliases

direct product

C₂×#851176

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2846	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

130947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232281, C₂×#72232282, C₂×#72232336, …
maximal normal subgroup of	C₂×#72232281, C₂×#72232282, C₂×#72232336, …

GAP

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

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

C2×#851176 #72337964