#7081514

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#24440	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=h,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285261
maximal normal subgroup of	#72285261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6988710);