#6730918

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285539, #72285584
maximal normal subgroup of	#72285539, #72285584

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6638114);