#288808

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,13)(8,18)(9,19)(10,12)(11,16)(14,15)(17,20)(21,22,25,29,24,28,31,27)(23,30,34,36,33,26,32,35);(1,2)(3,4)(5,7,15,8,10,17,9,16)(6,11,19,20,12,18,14,13)(21,23,31,32,24,33,25,34)(22,26,27,36,28,30,29,35);(2,4)(6,12)(11,18)(13,20)(14,19)(22,27)(23,32)(25,31)(26,30)(28,29)(33,34);(5,8,15,17,10,16,9,7)(6,13,19,11,12,20,14,18)(22,28)(26,30)(27,29)(35,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,d^2=ehi,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, 196004);