#7524623

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1514 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(18,22)(19,26)(25,30)(28,31);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,27)(20,22)(21,30)(23,26)(24,28)(25,29)(31,32);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(17,19,20,26)(18,23,22,27)(21,28,29,31)(24,30,32,25);(10,14)(13,16)(17,19)(18,24)(20,26)(21,31)(22,32)(23,25)(27,30)(28,29);(2,7)(6,8)(19,26)(23,27)(24,32)(28,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7431819);