#7019774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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	#59004	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1937 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)(17,18,21,25)(19,26,27,32)(20,23,28,24)(22,30,31,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(19,27)(22,31)(26,32)(29,30);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,28,30,21,27,20,29)(18,22,23,32,25,31,24,26);(2,5)(6,8)(11,16)(13,15)(17,20,21,28)(18,23,25,24)(19,29,27,30)(22,32,31,26);(10,14)(13,15)(17,20,21,28)(18,24,25,23)(19,30,27,29)(22,32,31,26)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6926970);