C₂×#2078612 #72325601

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

Aliases

direct product

C₂×#2078612

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

39023 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)(14,16)(17,18)(19,20)(21,25)(22,23)(24,26);(7,8,9,10)(11,13)(12,15)(14,18)(16,17)(20,23)(24,25);(4,6)(7,8)(9,10)(12,16)(15,17)(19,20,22,23)(21,25,26,24);(1,2);(8,10)(11,12)(13,17)(14,16)(15,18)(19,20)(21,24)(22,23)(25,26);(3,4)(5,6)(7,8)(9,10)(11,13)(12,17)(14,18)(15,16)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232183, C₂×#72232090, C₂×#72232089
maximal normal subgroup of	C₂×#72232183, C₂×#72232090, C₂×#72232089

GAP

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

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

C2×#2078612 #72325601