C₂²×#2760366 #72297008

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄²

Aliases

direct product

C₂²×#2760366

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 7638 conjugacy classes of subgroups; only the 38096 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 28 points:

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

Conjugacy classes & character table

296 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)(10,11)(22,23)(26,27), (5,6)(7,8)(9,10,12,11)(15,19)(18,20)(21,22,25,23)(24,26,28,27), (5,6,7,8)(9,10,12,11)(14,17)(15,19)(22,23)(24,28), (9,10,12,11)(13,14)(15,20)(16,17)(18,19)(21,23,25,22)(24,27,28,26), (5,6)(7,8)(9,11,12,10)(13,15,16,19)(14,18,17,20)(21,24,25,28)(22,26,23,27), (3,4), (1,2)(3,4));

C22×#2760366 #72297008