#480704

order	512
nilpotency class	4
isoclinism	isoclinic class of #480690
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×C₄⋊C₄	C₂⁴
prime radical	C₂×#537	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,22,28,23,29,19,25)(20,32,21,24,31,26,30,27);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(18,24)(19,22)(21,30)(25,32)(26,28)(27,29);(3,9)(6,12)(10,15)(13,16)(17,19,23,22)(18,25,29,28)(20,21,31,30)(24,26,27,32);(2,7)(6,12)(8,14)(13,16)(17,20,23,31)(18,26,29,32)(19,30,22,21)(24,28,27,25)

Polycyclic presentation:

2^9|a^2=f,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,e^b=ei,f^b=fgi,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, 387900);