#480689

order	512
nilpotency class	4
isoclinism	isoclinic class of #480684
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×C₄⋊C₄	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

877 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387885);