#482167

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

883 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,28,31,24)(20,25)(21,26,35,23)(22,27)(29,41,33,38)(30,36)(32,42)(34,40,45,37)(39,44)(43,47)(46,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,23)(20,32)(21,35)(24,36)(25,40)(26,27)(28,39)(29,33)(30,44)(37,42)(38,48)(41,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19)(18,24)(20,33)(21,30)(22,31)(23,36)(25,41)(26,39)(27,28)(29,32)(34,46)(35,44)(37,48)(38,42)(40,47)(43,45);(17,20,22,32)(18,25,27,42)(19,29,31,33)(21,34,35,45)(23,37,26,40)(24,38,28,41)(30,43,44,46)(36,47,39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389363);