C₂×#2131808 #72294594

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

Aliases

direct product

C₂×#2131808

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

28109 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293940
maximal normal subgroup of	C₂×#72293940
maximal quotient of	C₂×#72293964

GAP

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

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

C2×#2131808 #72294594