#4599186

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2006 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)(5,6,9,10)(7,12,11,8)(13,14,18,19)(15,20,16,17)(21,22,26,30)(23,31,33,27)(24,28,32,36)(25,34,35,29);(6,10)(7,11)(14,19)(15,16)(21,23,26,33)(22,27,30,31)(24,25,32,35)(28,29,36,34);(6,10)(8,12)(13,15,18,16)(14,17,19,20)(21,24)(22,28)(23,25)(26,32)(27,29)(30,36)(31,34)(33,35);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,16,18,15)(14,17,19,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,33)(32,36)(34,35);(5,8)(6,11)(7,10)(9,12)(13,17)(14,15)(16,19)(18,20)(21,25)(22,29)(23,32)(24,33)(26,35)(27,36)(28,31)(30,34)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=h,d^2=g,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, 4506382);