#7011368

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
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	#30396	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1559 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6918564);