C₂×#815832 #72338012

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

Aliases

direct product

C₂×#815832

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

38245 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232287, C₂×#72232297, C₂×#72232298
maximal normal subgroup of	C₂×#72232287, C₂×#72232297, C₂×#72232298

GAP

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

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

C2×#815832 #72338012