#517541

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2374 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=i,d^2=fh,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 424737);