#4668243

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1678 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)(5,6,8,10)(7,11,12,9)(14,17)(15,16)(21,22)(23,30)(24,25)(26,27)(28,31)(29,34)(32,33)(35,36);(5,6,8,10)(7,9,12,11)(13,14,18,17)(15,19,16,20)(22,27)(23,31)(24,34)(33,35);(5,6)(7,9)(8,10)(11,12)(13,15,18,16)(14,19,17,20)(21,23)(22,28)(24,32)(25,35)(26,31)(27,30)(29,33)(34,36);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,15)(14,19,17,20)(21,24)(22,29)(23,32)(25,27)(26,34)(28,33)(30,35)(31,36);(5,7)(6,9)(8,12)(10,11)(13,17,18,14)(15,19,16,20)(21,25,26,29)(22,24,27,34)(23,33,31,35)(28,36,30,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=gh,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=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, 4575439);