#747571

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2006 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 654767);