#5034875

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 6291456
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2677 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)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18)(19,22)(20,23)(21,24)(26,29)(27,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,30)(28,29)(31,32);(2,5)(6,7)(10,14)(12,16)(17,19)(18,22)(20,24)(21,23)(25,26,28,29)(27,32,31,30);(1,3)(2,6)(4,8)(5,7)(9,12,13,16)(10,15,14,11)(17,19)(18,22)(20,24)(21,23)(25,27,28,31)(26,30,29,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,23)(22,24)(25,26)(27,32)(28,29)(30,31)

Polycyclic presentation:

2^9|c^2=i,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266509, #72266523, #72266565, …
maximal normal subgroup of	#72266509, #72266523, #72266565, …
maximal quotient of	#72334891

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4942071);