#4308445

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 134217728
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

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

2818 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215641);