C₂×#157268 #72295010

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂×#157268

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

34165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293962, C₂×#72293964, C₂×#72293986
maximal normal subgroup of	C₂×#72293962, C₂×#72293964, C₂×#72293986

GAP

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

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

C2×#157268 #72295010