C₂×#540715 #72325228

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

Aliases

direct product

C₂×#540715

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

42125 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)(11,12)(13,16)(14,15)(17,18)(19,20,23,24)(21,25,26,22);(3,4,6,5)(7,8)(9,10)(12,15)(16,17)(20,24)(21,26);(7,8)(9,10)(20,24)(22,25);(7,8,9,10)(11,12)(13,17)(14,15)(16,18)(19,21)(20,22)(23,26)(24,25);(1,2);(3,5,6,4)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,22)(20,21)(23,25)(24,26)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232093, C₂×#72232145, C₂×#72232136
maximal normal subgroup of	C₂×#72232093, C₂×#72232145, C₂×#72232136

GAP

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

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

C2×#540715 #72325228