#556106

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 463302);