C₂×#2077471 #72332884

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

Aliases

direct product

C₂×#2077471

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

30655 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323415, C₂×#72323408, C₂×#72323401
maximal normal subgroup of	C₂×#72323415, C₂×#72323408, C₂×#72323401

GAP

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

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

C2×#2077471 #72332884