#5083056

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
automorphism	Aut(G) order 6291456
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2891 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)(13,14,17,18)(15,16,19,20)(21,22)(23,27)(24,26)(25,28)(29,35)(30,31)(32,34)(33,36);(5,6,9,7)(8,11,12,10)(14,18)(15,19)(22,24)(27,31)(28,32)(35,36);(6,7)(10,11)(13,15,17,19)(14,16,18,20)(21,23,26,30)(22,27,24,31)(25,33,34,29)(28,36,32,35);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,19)(18,20)(21,24,26,22)(23,27,30,31)(25,32,34,28)(29,35,33,36);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,20)(18,19)(21,25,26,34)(22,28,24,32)(23,29,30,33)(27,35,31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 4990252);