#545420

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	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=g,e^2=g,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, 452616);