C₂×#1420788 #72337904

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

Aliases

direct product

C₂×#1420788

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

36785 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)(9,13)(10,12)(11,14)(16,20)(17,22)(23,24,27,25)(26,28,30,29);(7,8,10,12)(9,11,14,13)(15,16)(17,21)(18,22)(19,20);(3,4)(5,6)(9,14)(11,13)(16,20)(17,22)(23,24,27,25)(26,28,30,29);(4,5)(7,8)(9,13)(10,12)(11,14)(15,17,19,22)(16,21,20,18)(23,25,27,24)(26,28,30,29);(1,2);(3,5,6,4)(7,9)(8,11)(10,14)(12,13)(15,18,19,21)(16,17,20,22)(23,26,27,30)(24,28,25,29)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232357, C₂×#72232334, C₂×#72232336
maximal normal subgroup of	C₂×#72232357, C₂×#72232334, C₂×#72232336

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

C2×#1420788 #72337904