#7019004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1498 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,23,27,21,25)(19,24,29,20,30,28,32,31);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,29)(21,26)(22,25)(23,27)(24,30)(31,32);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,27,28)(20,25,31,26)(21,32,22,29);(11,16)(13,15)(17,20,22,28,23,31,21,24)(18,19,26,29,27,30,25,32);(10,14)(13,15)(17,21,23,22)(18,25,27,26)(19,29,30,32)(20,28,31,24)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6926200);