#7598679

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#59824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1729 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fhi,d^2=fg,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,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, 7505875);