#552801

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2678 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:

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

Polycyclic presentation:

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

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, 459997);