#4611563

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=fg,e^2=gh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4518759);