#7598705

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518904
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	#59654	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1818 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)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,27,29,24)(20,23,25,21)(28,32,30,31);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,24,26,27)(20,31,25,32)(21,28,23,30);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,21,18,22,25,23,26)(19,24,30,31,29,27,28,32);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(17,20,21,18,22,25,23,26)(19,27,30,32,29,24,28,31);(1,4,5,3)(2,7,8,6)(17,21,22,23)(18,25,26,20)(19,28,29,30)(24,32,27,31)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7505901);