C₂×#1508994 #72338010

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂×#1508994

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35241 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232286, C₂×#72232297, C₂×#72232330
maximal normal subgroup of	C₂×#72232286, C₂×#72232297, C₂×#72232330

GAP

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

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

C2×#1508994 #72338010