C₂×#690272 #72325338

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#690272

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

40431 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)(11,12,15,18)(13,16,17,14)(19,20,25,24)(21,22,26,23);(3,4,5,6)(7,8)(9,10)(19,21,25,26)(20,22,24,23);(4,6)(11,13,15,17)(12,16,18,14)(19,22,25,23)(20,26,24,21);(4,6)(7,8,9,10)(11,13)(12,16)(14,18)(15,17)(19,23,25,22)(20,26,24,21);(1,2);(3,4)(5,6)(7,8)(9,10)(11,14,15,16)(12,17,18,13)(19,24,25,20)(21,22,26,23)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232187, C₂×#72232131, C₂×#72232128
maximal normal subgroup of	C₂×#72232187, C₂×#72232131, C₂×#72232128

GAP

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

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

C2×#690272 #72325338