#480700

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

896 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)(19,27)(20,24)(21,34,35,25)(22,26)(23,29,37,44)(28,41)(30,42)(31,45)(32,39)(33,48,47,40)(36,43,38,46);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19,22,30)(18,23,26,37)(20,31,32,28)(21,29,35,44)(24,38,39,36)(25,27,34,42)(33,46,47,43)(40,45,48,41);(3,9)(6,12)(10,15)(13,16)(20,32)(24,39)(28,31)(33,47)(36,38)(40,48)(41,45)(43,46);(2,7)(6,12)(8,14)(13,16)(17,20,22,32)(18,24,26,39)(19,28,30,31)(21,33,35,47)(23,36,37,38)(25,40,34,48)(27,41,42,45)(29,43,44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,d^b=di,f^b=fg,d^c=di,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, 387896);