#5025781

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 4194304
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2673 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)(6,9)(7,11)(14,17)(15,19)(21,22,25,28)(23,26,30,34)(24,31,32,27)(29,35,36,33);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,30)(22,26,28,34)(24,29,32,36)(27,33,31,35);(6,9)(10,12)(13,14,16,17)(15,20,19,18)(21,23)(22,26)(24,29)(25,30)(27,33)(28,34)(31,35)(32,36);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,14)(15,20)(16,17)(18,19)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(5,6)(7,12)(8,9)(10,11)(13,15)(14,18)(16,19)(17,20)(21,24,25,32)(22,27,28,31)(23,29,30,36)(26,33,34,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=g,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4932977);