#5481044

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2120 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)(8,11,12,10)(13,14,18,16)(15,20,17,19)(21,22,26,27)(23,24,29,32)(25,28,33,35)(30,31,34,36);(5,6,9,7)(8,11,12,10)(13,14,18,16)(15,19,17,20)(22,27)(23,29)(25,33)(31,36);(6,7)(10,11)(13,15,18,17)(14,19,16,20)(21,23)(22,24)(25,34)(26,29)(27,32)(28,36)(30,33)(31,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,19,17,20)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);(5,8)(6,10)(7,11)(9,12)(13,17,18,15)(14,19,16,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,35)(29,34)(32,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fgh,d^2=g,e^2=fgh,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, 5388240);