#5045316

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2188 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=f,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=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, 4952512);