#7600076

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30400	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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)(3,7)(4,6)(5,8)(17,18)(19,29)(20,28)(21,24)(22,26)(23,25)(27,30)(31,32);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,31,23,30,22,21)(18,24,26,27,25,32,28,29);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(18,25)(19,30)(21,31)(26,28);(2,6)(5,7)(10,14)(13,16)(17,20,23,22)(18,26,25,28)(19,31,30,21)(24,27,32,29);(2,6)(5,7)(17,21,22,30,23,31,20,19)(18,27,28,24,25,29,26,32)

Polycyclic presentation:

2^9|b^2=fi,c^2=h,d^2=i,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7507272);