#4721422

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	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=fghi,d^2=fghi,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	#72334169, #72334174, #72334236
maximal normal subgroup of	#72334169, #72334174, #72334236

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4628618);