#291742

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1063 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198938);