C₂²×#5021210 #72295286

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂²×#5021210

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

446348 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72293994
maximal normal subgroup of	C₂×#72293994

GAP

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

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

C22×#5021210 #72295286