#6495993

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#33185	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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)(11,15)(14,16)(17,18,23,26)(19,27,25,22)(20,29,21,24)(28,30,31,32);(9,10)(11,14)(12,13)(15,16)(17,19,22,18,23,25,27,26)(20,28,30,24,21,31,32,29);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20,23,21)(18,24,26,29)(19,28,25,31)(22,30,27,32);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,21,23,20)(18,24,26,29)(19,28,25,31)(22,32,27,30);(9,11,12,15)(10,14,13,16)(20,21)(24,29)(28,31)(30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,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, 6403189);