#284334

order	512
nilpotency class	3
isoclinism	isoclinic class of #264491
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

752 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)(7,14)(8,10)(9,11)(12,16)(13,17)(15,20)(18,19)(21,22,25,35,26,31,27,30)(23,29,34,36,32,28,24,33);(1,2)(3,4)(6,10)(7,15)(8,17)(11,13)(12,20)(14,19)(21,23,27,24,26,32,25,34)(22,28,30,36,31,29,35,33);(2,4)(6,11)(10,13)(12,19)(14,20)(21,24)(22,29)(23,27)(25,32)(26,34)(28,31)(30,36)(33,35);(5,7,9,15)(6,12,11,19)(8,18,17,16)(10,14,13,20)(21,25,26,27)(22,30,31,35)(23,24,32,34)(28,33,29,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,d^2=hi,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=dhi,e^b=eh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 191530);