#6458567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	#35272	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,14,12,16)(17,18,23,26,24,28,22,27)(19,29,30,32,31,21,25,20);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,20)(21,26)(22,30)(23,25)(24,31)(27,29)(28,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,23,29,24,32,22,21)(18,19,26,30,28,31,27,25);(1,4,5,3)(2,8,6,7)(17,21,23,20,24,29,22,32)(18,25,26,19,28,30,27,31);(17,22,24,23)(18,26,28,27)(19,30,31,25)(20,21,32,29)

Polycyclic presentation:

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

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, 6365763);