C₂×#525871 #72325282

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#525871

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

48089 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232083, C₂×#72232121, C₂×#72232161
maximal normal subgroup of	C₂×#72232083, C₂×#72232121, C₂×#72232161

GAP

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

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

C2×#525871 #72325282