#315088

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	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 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)(3,5)(4,6)(7,8)(9,10,13,11,14,16,12,15)(17,18,22,24)(19,25,29,32)(20,30,28,23)(21,31,27,26);(1,3,4,7)(2,5,6,8)(9,11,12,10,14,15,13,16)(17,19,27,20,22,29,21,28)(18,23,31,32,24,30,26,25);(2,6)(5,8)(10,11)(12,13)(15,16)(18,24)(23,30)(25,32)(26,31);(17,20,27,29,22,28,21,19)(18,25,31,23,24,32,26,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=ef,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fi,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, 222284);