C₂×#1162821 #72294737

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

Aliases

direct product

C₂×#1162821

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

32721 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)(15,16,19,21)(17,22,18,20)(23,24)(25,29)(26,27)(28,30);(7,9,11,14)(8,12,10,13)(15,17)(16,20)(18,19)(21,22)(24,27)(28,29);(4,6)(8,10)(12,13)(17,18)(20,22)(23,25,26,30)(24,28,27,29);(3,4)(5,6)(7,9)(8,12)(10,13)(11,14)(15,16)(17,22)(18,20)(19,21)(23,25)(24,29)(26,30)(27,28);(7,10,11,8)(9,13,14,12)(15,18,19,17)(16,20,21,22)(23,24)(25,29)(26,27)(28,30);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232681, C₂×#72231820
maximal normal subgroup of	C₂×#72232681, C₂×#72231820
maximal quotient of	C₂×#72293964, C₂×#72278160

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

C2×#1162821 #72294737