#4668107

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
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

2359 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)(7,9)(8,10)(11,12)(13,14,17,15)(16,18,20,19)(22,26)(23,29)(24,31)(35,36);(6,7)(10,11)(14,15)(18,19)(21,22,25,26)(23,27,29,33)(24,28,31,34)(30,35,32,36);(5,6,9,7)(8,11,12,10)(13,15,17,14)(16,18,20,19)(21,23)(22,27)(24,32)(25,29)(26,33)(28,36)(30,31)(34,35);(1,2,3,4)(5,7,9,6)(8,11,12,10)(21,22)(23,27)(24,28)(25,26)(29,33)(30,35)(31,34)(32,36);(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,24)(22,28)(23,30)(25,31)(26,34)(27,35)(29,32)(33,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=h,d^2=fhi,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.

Relationships

maximal subgroup of	#72304041, #72304047, #72304136
maximal normal subgroup of	#72304041, #72304047, #72304136

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4575303);