#254319

order	512
nilpotency class	3
isoclinism	isoclinic class of #254299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#120	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,g^c=gi,f^d=fh

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, 161515);