#224393

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

983 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)(11,15,16,13)(12,14)(17,18)(19,23)(20,28,29,32)(21,26)(22,25)(24,27,31,30);(1,2,4,5)(3,7,8,6)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,28,31,32);(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,28)(25,31)(26,32);(1,3,4,8)(2,6,5,7)(18,25)(23,26)(24,31)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=h,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,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, 131589);