#7160918

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	#45303	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1577 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7068114);