#758512

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

2042 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)(5,6,9,8)(7,11,12,10)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(14,17)(18,20)(22,26)(24,32)(27,29)(30,36);(1,2,3,4)(13,14)(15,18)(16,17)(19,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,7,9,12)(6,10,8,11)(13,15,16,19)(14,18,17,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,8,9,6)(7,11,12,10)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=fhi,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, 665708);