#6678982

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

1810 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)(11,13)(12,14)(15,16)(17,18,23,27)(19,21,28,24)(20,25,30,32)(22,26,31,29);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,25)(21,27)(22,26)(23,28)(29,31)(30,32);(10,14)(13,16)(17,20)(18,25)(19,22)(21,26)(23,30)(24,29)(27,32)(28,31);(1,3,4,7)(2,6,5,8)(17,21,23,24)(18,19,27,28)(20,29,30,26)(22,25,31,32);(2,5)(6,8)(17,22)(18,26)(19,20)(21,25)(23,31)(24,32)(27,29)(28,30)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,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.

Relationships

maximal subgroup of	#72257411, #72257520, #72264075
maximal normal subgroup of	#72257411, #72257520, #72264075
maximal quotient of	#72308550

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6586178);