C₂×#657241 #72338006

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#657241

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

36067 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232346, C₂×#72232297, C₂×#72232298
maximal normal subgroup of	C₂×#72232346, C₂×#72232297, C₂×#72232298

GAP

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

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

C2×#657241 #72338006