#301967

order	512
nilpotency class	3
isoclinism	isoclinic class of #301949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1122 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)(3,11)(4,10)(5,9)(6,8)(7,12)(13,16)(14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(1,2,6,10,5,9,4,8)(3,7,14,16,13,15,12,11)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,30,31,32);(1,2,5,9)(3,7,13,15)(4,8,6,10)(11,14,16,12)(17,20)(18,23)(19,26,27,28)(21,29)(22,30,25,32)(24,31);(1,3,4,12,5,13,6,14)(2,7,8,11,9,15,10,16)(18,24)(22,25)(23,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209163);