#7049317

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#31056	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fg,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6956513);