#6458871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,26,25,21)(20,24,29,31)(27,32,28,30);(9,11)(10,13)(12,15)(14,16)(17,19,21,18,22,25,26,23)(20,28,30,31,29,27,32,24);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(18,23)(19,25)(24,31)(27,28);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,29)(18,24,23,31)(19,27,25,28)(21,30,26,32);(20,29)(24,31)(27,28)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366067);