#7620311

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#59952	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(3,7)(4,8)(5,6)(10,14)(11,12)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,27)(18,23,24,30)(20,29,28,21)(25,32,31,26);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(18,24)(19,27)(20,28)(26,32);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,20,21,19,22,28,29,27)(18,25,26,23,24,31,32,30);(2,6)(7,8)(9,12,13,11)(10,15,14,16)(17,21,22,29)(18,26,24,32)(19,20,27,28)(23,25,30,31)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=fghi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7527507);