C₂×#7728255 #72324176

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7728255

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

87357 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 30 points:

(15,16)(17,24)(18,21)(19,22)(20,23)(25,30)(26,28)(27,29);(4,6)(9,13)(12,14)(19,27)(22,29)(25,28)(26,30);(3,4)(5,6)(15,17,20,21)(16,18,23,24)(19,25,27,28)(22,26,29,30);(8,11)(9,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30);(1,2);(7,8,10,11)(9,12,13,14)(15,18)(16,21)(17,23)(19,26)(20,24)(22,28)(25,29)(27,30);(7,9,10,13)(8,12,11,14)(15,19,20,27)(16,22,23,29)(17,25,21,28)(18,26,24,30)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72256612
maximal normal subgroup of	C₂×#72256612
maximal quotient of	C₂×#72323320, C₂×#72323322

GAP

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

gap> G := Group((15,16)(17,24)(18,21)(19,22)(20,23)(25,30)(26,28)(27,29), (4,6)(9,13)(12,14)(19,27)(22,29)(25,28)(26,30), (3,4)(5,6)(15,17,20,21)(16,18,23,24)(19,25,27,28)(22,26,29,30), (8,11)(9,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30), (1,2), (7,8,10,11)(9,12,13,14)(15,18)(16,21)(17,23)(19,26)(20,24)(22,28)(25,29)(27,30), (7,9,10,13)(8,12,11,14)(15,19,20,27)(16,22,23,29)(17,25,21,28)(18,26,24,30));

C2×#7728255 #72324176