#647561

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=h,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 554757);