#7590556

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#44909	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=g,d^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,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, 7497752);