#6720097

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2657	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6627293);