C₂×#636210 #72325242

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

Aliases

direct product

C₂×#636210

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2846	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

134875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232082, C₂×#72232106, C₂×#72232149
maximal normal subgroup of	C₂×#72232082, C₂×#72232106, C₂×#72232149

GAP

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

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

C2×#636210 #72325242