#6458578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#35528	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2021 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)(17,18,23,21,24,28,22,27)(19,29,30,32,31,20,26,25);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,21)(22,30)(23,26)(24,31)(27,29)(28,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,23,25,24,29,22,32)(18,26,21,19,28,30,27,31);(1,4,5,3)(2,8,6,7)(17,21,23,28,24,27,22,18)(19,25,30,29,31,32,26,20);(17,22,24,23)(18,27,28,21)(19,30,31,26)(20,25,29,32)

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,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=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, 6365774);