#7090816

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	#23857	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,26,19,23)(20,24,28,31,29,32,27,30);(9,10)(11,15)(12,14)(13,16)(18,23)(19,21)(20,28)(25,26)(27,29)(30,31);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,22,21)(18,23,26,25)(20,28,29,27)(24,31,32,30);(11,16)(13,15)(17,20,21,28,22,29,19,27)(18,24,25,31,26,32,23,30);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,19,22,21)(18,23,26,25)(20,28,29,27)(24,31,32,30)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=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, 6998012);