#4607659

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2136 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)(7,8)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)(22,25)(24,31)(27,33)(29,35);(6,10)(7,12)(14,15)(18,19)(21,22,26,25)(23,27,30,33)(24,28,31,32)(29,34,35,36);(6,10)(8,11)(13,14,17,15)(16,19,20,18)(21,23)(22,27)(24,29)(25,33)(26,30)(28,34)(31,35)(32,36);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,15,17,14)(16,19,20,18)(21,24)(22,28)(23,29)(25,32)(26,31)(27,34)(30,35)(33,36);(5,8,9,11)(6,12,10,7)(13,16,17,20)(14,18,15,19)(21,25,26,22)(23,27,30,33)(24,32,31,28)(29,34,35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=h,d^2=g,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, 4514855);