#6495193

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35278	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402389);