#232154

order	512
nilpotency class	3
isoclinism	isoclinic class of #232153
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3184	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

433 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8,11,7)(4,12,14,16)(5,9)(6,10)(13,15)(17,18)(19,29)(20,26,36,25)(21,37,38,45)(22,39)(23,28)(24,32)(27,35)(30,43,47,44)(31,41,34,48)(33,46)(40,42);(1,3)(2,7)(4,13)(5,14)(6,11)(8,10)(9,12)(15,16)(17,19)(18,24)(20,34)(21,30)(22,40)(23,33)(25,43)(26,44)(27,46)(28,42)(29,39)(31,36)(32,35)(37,48)(38,47)(41,45);(1,4,13,3,6,14,5,11)(2,8,9,12,10,7,15,16)(17,20,35,38,23,36,22,21)(18,25,27,37,28,26,39,45)(19,30,40,31,33,47,32,34)(24,41,29,44,42,48,46,43);(2,9,10,15)(7,16,8,12)(17,21,22,36,23,38,35,20)(18,26,27,45,28,25,39,37)(19,31,32,30,33,34,40,47)(24,41,29,44,42,48,46,43)

Polycyclic presentation:

2^9|a^2=g,c^2=efh,d^2=e,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cef,d^b=de,e^b=ei,f^b=fhi,g^b=ghi,g^c=gh,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139350);