#333263

order	512
nilpotency class	3
isoclinism	isoclinic class of #333199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19701	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

626 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:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240459);