#6524516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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)(3,6)(4,8)(5,7)(17,18,23,27,22,28,21,26)(19,29,30,24,25,32,31,20);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,30)(22,25)(23,31)(26,32)(27,29);(2,7)(6,8)(10,13)(11,15)(17,20,22,24)(18,25,28,19)(21,32,23,29)(26,30,27,31);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,28,27)(19,30,25,31)(20,29,24,32);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(17,21,22,23)(18,27,28,26)(19,31,25,30)(20,29,24,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6431712);