#4416506

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=fi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4323702);