C₂×#612907 #72334690

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

Aliases

direct product

C₂×#612907

Subgroups

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

67141 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)(8,10)(11,12,15,13)(14,16,18,17)(19,20)(21,22)(23,24)(25,26);(4,6)(20,24)(22,26);(8,10)(11,12,15,13)(14,16,18,17)(19,21)(20,22)(23,25)(24,26);(3,4)(5,6)(7,8,9,10)(11,13,15,12)(14,16,18,17)(19,22)(20,25)(21,24)(23,26);(1,2);(11,14)(12,16)(13,17)(15,18)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297882, C₂×#72309280
maximal normal subgroup of	C₂×#72297882, C₂×#72309280
maximal quotient of	C₂×#72334407, C₂×#72298504

GAP

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

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

C2×#612907 #72334690