#698300

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1918 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)(5,6,9,8)(7,10,11,12)(15,18)(16,20)(21,22,26,25)(23,27,29,31)(24,28,34,33)(30,35,32,36);(7,11)(10,12)(13,14,17,19)(15,16,18,20)(22,25)(23,29)(28,33)(30,32);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,15,17,18)(14,16,19,20)(21,23)(22,27)(24,32)(25,31)(26,29)(28,36)(30,34)(33,35);(5,8,9,6)(7,12,11,10)(13,16)(14,18)(15,19)(17,20)(21,24,26,34)(22,28,25,33)(23,30,29,32)(27,35,31,36);(21,25,26,22)(23,31,29,27)(24,33,34,28)(30,36,32,35)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=h,e^2=f,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, 605496);