#6489089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	#32904	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1774 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)(11,14)(12,15)(13,16)(18,23)(21,29)(24,31)(25,32)(26,30)(27,28);(11,12)(14,15)(17,18,22,26)(19,24,20,25)(21,30,29,23)(27,32,28,31);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,19,22,20)(18,24,26,25)(21,27,29,28)(23,31,30,32);(9,12,13,11)(10,15,16,14)(17,20,22,19)(18,25,26,24)(21,28,29,27)(23,32,30,31);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(19,20)(24,25)(27,28)(31,32)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=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, 6396285);