#5610141

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
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

1866 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)(5,6,9,7)(8,10,12,11)(15,19)(18,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(8,12)(10,11)(14,17)(18,20)(21,22,25,28)(23,26,31,29)(24,27,33,36)(30,34,32,35);(6,7)(10,11)(13,14,16,17)(15,18,19,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,23)(22,26)(24,32)(25,31)(27,35)(28,29)(30,33)(34,36);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24,25,33)(22,27,28,36)(23,30,31,32)(26,34,29,35)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5517337);