#7047671

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6954867);