C₂×#606390 #72294543

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#606390

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

30237 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)(9,10)(11,13)(12,14)(15,16,19,22)(17,18,21,20)(23,24,27,30)(25,26,29,28);(7,9,11,14)(8,12,13,10)(15,17)(16,20)(18,22)(19,21)(23,24,27,30)(25,26,29,28);(3,4,5,6)(8,13)(10,12)(15,17,19,21)(16,20,22,18)(23,25)(24,28)(26,30)(27,29);(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22)(23,25)(24,26)(27,29)(28,30);(7,9)(8,10)(11,14)(12,13)(15,18)(16,21)(17,22)(19,20)(23,26)(24,29)(25,30)(27,28);(1,2)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293937
maximal normal subgroup of	C₂×#72293937
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)(9,10)(11,13)(12,14)(15,16,19,22)(17,18,21,20)(23,24,27,30)(25,26,29,28), (7,9,11,14)(8,12,13,10)(15,17)(16,20)(18,22)(19,21)(23,24,27,30)(25,26,29,28), (3,4,5,6)(8,13)(10,12)(15,17,19,21)(16,20,22,18)(23,25)(24,28)(26,30)(27,29), (7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22)(23,25)(24,26)(27,29)(28,30), (7,9)(8,10)(11,14)(12,13)(15,18)(16,21)(17,22)(19,20)(23,26)(24,29)(25,30)(27,28), (1,2));

C2×#606390 #72294543