C₂×#613018 #72339989

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#613018

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

67165 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339991, #72339993, C₂×#72337223
maximal normal subgroup of	#72339991, #72339993, C₂×#72337223
maximal quotient of	C₂×#72337369

GAP

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

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

C2×#613018 #72339989