#7027568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#44873	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1529 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)(9,10)(11,13)(12,14)(15,16)(19,26)(20,21)(23,31)(24,30)(27,32)(28,29);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,30)(21,24)(22,25)(23,29)(26,32)(28,31);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,26,22,29,20,28)(18,23,24,31,25,27,30,32);(3,8)(6,7)(11,16)(13,15)(17,18)(19,27)(20,30)(21,24)(22,25)(23,29)(26,32)(28,31);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,20,22,21)(18,24,25,30)(19,28,29,26)(23,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6934764);