C₂×#579776 #72297002

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

Aliases

direct product

C₂×#579776

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2535	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

27325 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278602
maximal normal subgroup of	C₂×#72278602
maximal quotient of	C₂×#72278676

GAP

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

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

C2×#579776 #72297002