#481303

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

774 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388499);