C₂×#1517182 #72337877

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#1517182

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

39481 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232354, C₂×#72232308, C₂×#72232297
maximal normal subgroup of	C₂×#72232354, C₂×#72232308, C₂×#72232297

GAP

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

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

C2×#1517182 #72337877