C₂×#521329 #72331997

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#521329

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

38797 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323320, C₂×#72323322, C₂×#72323336
maximal normal subgroup of	C₂×#72323320, C₂×#72323322, C₂×#72323336

GAP

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

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

C2×#521329 #72331997