#480531

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
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	#35563	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,4,8,5,6,3,7)(9,10,13,17,14,16,21,23)(11,18,19,24,12,15,20,22);(2,6)(3,4)(9,11,14,12)(10,15,16,18)(13,19,21,20)(17,24,23,22);(1,3,5,4)(2,7,6,8)(11,12)(15,18)(19,20)(22,24);(9,12,14,11)(10,15,16,18)(13,20,21,19)(17,22,23,24)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ch,f^b=fg,d^c=di,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, 387727);