#6559078

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,15)(8,17)(9,13)(10,12)(11,14)(16,19)(18,20)(21,22)(23,30)(24,28)(25,34)(26,29)(27,32)(31,35)(33,36);(5,7,11,10)(6,12,14,15)(8,18,9,16)(13,20,17,19)(21,23,25,32,26,33,24,31)(22,27,28,30,29,35,34,36);(1,2,3,4)(5,8,11,9)(6,13,14,17)(7,16,10,18)(12,19,15,20)(21,22)(23,27)(24,34)(25,28)(26,29)(30,32)(31,36)(33,35);(5,9,11,8)(6,13,14,17)(7,16,10,18)(12,20,15,19)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,30,35,36);(5,10,11,7)(6,12,14,15)(8,18,9,16)(13,19,17,20)

Polycyclic presentation:

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

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, 6466274);