C₂²×#1427668 #72296982

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#1427668

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂²×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6189 conjugacy classes of subgroups; only the 36856 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:

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

Conjugacy classes & character table

368 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((5,6)(7,8)(18,21)(22,24)(25,26)(27,31)(28,29)(30,32), (6,7)(19,23)(22,24), (6,7)(10,13)(14,16)(18,21)(22,24)(26,29)(27,32), (6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,23)(25,26)(27,30)(28,29)(31,32), (5,7,8,6)(9,11,12,15)(10,14,13,16)(17,19,20,23)(18,22,21,24)(25,27,28,32)(26,30,29,31), (3,4), (1,2)(3,4));

C22×#1427668 #72296982