#6700324

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1809 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=fg,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	#72308707, #72308985
maximal normal subgroup of	#72308707, #72308985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6607520);