#7126729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1311 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=hi,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^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7033925);