C₂²×#2037594 #72296965

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

Aliases

direct product

C₂²×#2037594

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 6477 conjugacy classes of subgroups; only the 36168 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)(18,19)(22,23)(27,31)(30,32);(5,6)(7,8)(17,18)(19,21)(20,22)(23,24)(25,26)(27,32)(28,29)(30,31);(5,6,7,8)(10,13)(11,15)(18,19)(20,24)(26,29)(30,32);(5,6,7,8)(9,10)(11,16)(12,13)(14,15)(17,19,21,18)(20,22,24,23)(25,27,28,31)(26,30,29,32);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,32)(28,29)(30,31);(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	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)(18,19)(22,23)(27,31)(30,32), (5,6)(7,8)(17,18)(19,21)(20,22)(23,24)(25,26)(27,32)(28,29)(30,31), (5,6,7,8)(10,13)(11,15)(18,19)(20,24)(26,29)(30,32), (5,6,7,8)(9,10)(11,16)(12,13)(14,15)(17,19,21,18)(20,22,24,23)(25,27,28,31)(26,30,29,32), (5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,32)(28,29)(30,31), (3,4), (1,2)(3,4));

C22×#2037594 #72296965