#4308702

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 50331648
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=h,e^2=i,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, 4215898);