C₂×#690320 #72325336

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

Aliases

direct product

C₂×#690320

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

41109 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)(12,15)(16,17)(21,25)(23,26);(3,4,5,6)(7,8)(9,10)(11,12,14,15)(13,17,18,16);(4,6)(12,15)(13,18)(19,20,22,24)(21,23,25,26);(4,6)(7,8,9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,26)(22,24)(23,25);(1,2);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17)(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×#72232187, C₂×#72232135, C₂×#72232124
maximal normal subgroup of	C₂×#72232187, C₂×#72232135, C₂×#72232124

GAP

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

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

C2×#690320 #72325336