#7616319

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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)(4,5)(6,8)(11,12)(14,15)(17,18,22,23)(19,21,26,32)(20,27,30,24)(25,29,28,31);(2,5)(6,7)(9,10,13,16)(11,15,12,14)(17,19,22,26)(18,21,23,32)(20,28,30,25)(24,29,27,31);(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,25)(21,31)(22,30)(23,27)(26,28)(29,32);(9,12,13,11)(10,15,16,14)(17,21,23,26,22,32,18,19)(20,29,27,25,30,31,24,28);(1,3,4,8)(2,6,5,7)(20,30)(24,27)(25,28)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,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, 7523515);