#6579962

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#32913	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284905
maximal normal subgroup of	#72284905

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6487158);