C₂×#582895 #72294720

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂×#582895

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

31721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293944
maximal normal subgroup of	C₂×#72293944
maximal quotient of	C₂×#72293980

GAP

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

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

C2×#582895 #72294720