C₂²×#5473828 #72296949

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁶×C₄²

Aliases

direct product

C₂²×#5473828

Subgroups

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

At least 6927 conjugacy classes of subgroups; only the 35304 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 32 points:

(6,8)(18,21)(22,23)(27,31)(30,32);(5,6,7,8)(11,15)(14,16)(17,18,20,21)(19,23,24,22)(25,26,28,29)(27,32,31,30);(5,6,7,8)(10,13)(11,15)(18,21)(19,24)(26,29)(30,32);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,29)(31,32);(3,4);(1,2)(3,4)

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#72231349
maximal normal subgroup of	C₂²×#72231349

GAP

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

gap> G := Group((6,8)(18,21)(22,23)(27,31)(30,32), (5,6,7,8)(11,15)(14,16)(17,18,20,21)(19,23,24,22)(25,26,28,29)(27,32,31,30), (5,6,7,8)(10,13)(11,15)(18,21)(19,24)(26,29)(30,32), (5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32), (5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,29)(31,32), (3,4), (1,2)(3,4));

C22×#5473828 #72296949