C₂×#8598551 #72241209

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#8598551

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 10158 conjugacy classes of subgroups; only the 29549 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232845, C₂×#72232068, C₂×#72232275
maximal normal subgroup of	C₂×#72232845, C₂×#72232068, C₂×#72232275
maximal quotient of	C₂×#72232097

GAP

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

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

C2×#8598551 #72241209