C₂×#848381 #72337912

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#848381

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

45221 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)(9,11)(10,13)(12,14);(3,4,6,5)(8,9)(10,14)(15,16,18,20)(17,19,21,22)(24,27)(25,29);(4,5)(8,9)(10,14)(17,21)(19,22)(23,24,26,27)(25,30,29,28);(4,5)(7,9,11,8)(10,12,14,13)(15,16)(17,22)(18,20)(19,21)(23,24)(25,30)(26,27)(28,29);(1,2);(3,5,6,4)(7,10)(8,12)(9,13)(11,14)(15,17)(16,19)(18,21)(20,22)(23,25,26,29)(24,28,27,30)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232360, C₂×#72232339, C₂×#72232297
maximal normal subgroup of	C₂×#72232360, C₂×#72232339, C₂×#72232297

GAP

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

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

C2×#848381 #72337912