C₂×#3664121 #72294136

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

Aliases

direct product

C₂×#3664121

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

34819 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)(9,14,10,12)(15,16)(17,19)(18,20)(21,22)(23,24,28,29)(25,30,26,27);(7,9)(8,12)(10,11)(13,14)(15,17,18,21)(16,19,20,22)(25,26)(27,30);(4,6)(9,10)(12,14)(16,20)(17,21)(23,25,28,26)(24,27,29,30);(3,4)(5,6)(7,10,11,9)(8,12,13,14)(15,16)(17,19)(18,20)(21,22)(23,26,28,25)(24,27,29,30);(7,8)(9,12)(10,14)(11,13)(15,17)(16,19)(18,21)(20,22)(23,27)(24,26)(25,29)(28,30);(1,2)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293904
maximal normal subgroup of	C₂×#72293904
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((7,8,11,13)(9,14,10,12)(15,16)(17,19)(18,20)(21,22)(23,24,28,29)(25,30,26,27), (7,9)(8,12)(10,11)(13,14)(15,17,18,21)(16,19,20,22)(25,26)(27,30), (4,6)(9,10)(12,14)(16,20)(17,21)(23,25,28,26)(24,27,29,30), (3,4)(5,6)(7,10,11,9)(8,12,13,14)(15,16)(17,19)(18,20)(21,22)(23,26,28,25)(24,27,29,30), (7,8)(9,12)(10,14)(11,13)(15,17)(16,19)(18,21)(20,22)(23,27)(24,26)(25,29)(28,30), (1,2));

C2×#3664121 #72294136