#6476744

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2141 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(18,23)(20,21)(24,30)(25,31)(26,32)(27,29);(9,11)(10,13)(12,15)(14,16)(17,18,22,24)(19,26,28,25)(20,23,21,30)(27,31,29,32);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(18,24)(23,30)(25,26)(31,32);(11,15)(13,16)(17,19)(18,25)(20,29)(21,27)(22,28)(23,31)(24,26)(30,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,21)(18,23,24,30)(19,27,28,29)(25,32,26,31)

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=gi,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^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, 6383940);