C₂×#917297 #72332598

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

Aliases

direct product

C₂×#917297

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

46449 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)(11,12,14,16)(13,15,18,17)(19,20,22,25)(21,24,26,23);(3,4)(5,6)(7,8)(9,10)(11,12,14,16)(13,17,18,15)(19,21,22,26)(20,23,25,24);(19,21)(20,24)(22,26)(23,25);(1,2);(4,6)(11,12)(13,17)(14,16)(15,18);(3,4)(5,6)(11,13)(12,15)(14,18)(16,17)(19,20)(21,24)(22,25)(23,26)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323385, C₂×#72323389, C₂×#72323410
maximal normal subgroup of	C₂×#72323385, C₂×#72323389, C₂×#72323410

GAP

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

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

C2×#917297 #72332598