#326112

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25392	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=egh,c^2=e,d^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^c=gi

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, 233308);