#224410

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

975 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)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,25)(20,28,30,23)(21,24)(22,27)(26,31,29,32);(1,2,4,5)(3,7,8,6)(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31);(10,15)(11,12)(14,16)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29);(1,3,4,8)(2,6,5,7)(10,16)(14,15)(19,27)(22,25)(26,29)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=ehi,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131606);