C₂×#619125 #72338034

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#619125

Subgroups

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

54865 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)(21,25)(23,26);(3,4)(5,6)(7,8)(9,10)(11,12,14,15)(13,16,18,17);(4,6)(8,10)(12,15)(16,17)(19,20,22,24)(21,26,25,23);(1,2);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,20)(21,23)(22,24)(25,26);(3,4)(5,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,18)(19,21,22,25)(20,23,24,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232346, C₂×#72232286, C₂×#72210496
maximal normal subgroup of	C₂×#72232346, C₂×#72232286, C₂×#72210496

GAP

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

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

C2×#619125 #72338034