#7608105

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=ghi,d^2=g,e^2=fg,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,e^d=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, 7515301);