#4607208

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2086 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)(5,6,8,10)(7,9,11,12)(14,17)(18,20)(21,22,26,27)(23,24,31,28)(25,33,35,29)(30,36,34,32);(5,7,8,11)(6,9,10,12)(15,19)(18,20)(22,27)(24,28)(29,33)(32,36);(5,7)(6,9)(8,11)(10,12)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,27)(25,34)(26,31)(29,32)(30,35)(33,36);(1,2)(3,4)(13,15)(14,18)(16,19)(17,20)(21,24,26,28)(22,23,27,31)(25,32,35,36)(29,30,33,34);(5,6)(7,9)(8,10)(11,12)(21,25,26,35)(22,29,27,33)(23,30,31,34)(24,32,28,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=fg,d^2=f,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, 4514404);