#5474678

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1896 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)(6,7)(8,12)(13,14,16,18)(15,17,19,20)(21,22,26,27)(23,24,30,28)(25,29,34,35)(31,32,33,36);(5,6,9,7)(8,11,12,10)(15,19)(17,20)(22,27)(23,30)(29,35)(31,33);(8,12)(10,11)(13,15)(14,17)(16,19)(18,20)(21,23,26,30)(22,28,27,24)(25,33,34,31)(29,32,35,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,24,26,28)(22,23,27,30)(25,32,34,36)(29,31,35,33);(5,8,9,12)(6,10,7,11)(13,14)(15,17)(16,18)(19,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,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, 5381874);