#7598087

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#59419	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1640 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)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,25)(21,26)(22,30)(23,29)(24,28)(31,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,23)(21,22)(24,31)(26,28)(27,29)(30,32);(10,13)(14,15)(17,20,24,31,23,19,22,21)(18,26,30,27,29,32,28,25);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,22,19,23,31,24,20)(18,27,28,26,29,25,30,32);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,30)(19,21,20,31)(25,26,27,32)

Polycyclic presentation:

2^9|c^2=h,d^2=fgi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7505283);