#7610792

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1489 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,23)(21,25)(22,27)(24,30)(26,32)(29,31);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,32)(26,31);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,23,18)(19,24,30,28)(21,25,31,29)(22,26,32,27);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,22,24,23,31,32,28)(18,25,27,30,20,29,26,19);(2,8)(6,7)(17,22,23,32)(18,26,20,27)(19,29,30,25)(21,24,31,28)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7517988);