#545425

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2654 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 452621);