C₂²×#580926 #72297009

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#580926

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂³×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

433752 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:

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

Conjugacy classes & character table

416 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((11,15)(14,16)(20,24)(22,23)(26,28)(30,31), (5,6,8,7)(11,15)(14,16)(17,18,21,19)(20,22,24,23)(25,26,29,28)(27,30,32,31), (5,6,8,7)(10,13)(11,15)(18,19)(20,24)(26,28)(30,31), (5,6,8,7)(9,10)(11,16)(12,13)(14,15)(17,19,21,18)(20,23,24,22)(25,27,29,32)(26,30,28,31), (5,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,31,32,30), (3,4), (1,2)(3,4));

C22×#580926 #72297009