#482571

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284138, #72284183, #72284337, …
maximal normal subgroup of	#72284138, #72284183, #72284337, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389767);