#216521

order	512
nilpotency class	3
isoclinism	isoclinic class of #162662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=ehi,f^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 123717);