#6458504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36624	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2216 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365700);