C₂×#3071666 #72294526

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

Aliases

direct product

C₂×#3071666

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

31363 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293912
maximal normal subgroup of	C₂×#72293912
maximal quotient of	C₂×#72293943

GAP

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

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

C2×#3071666 #72294526