#6606019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574031
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,31)(21,25)(22,27)(23,29)(26,28)(30,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,28,20,22,30,21,29)(18,23,25,32,27,31,26,24);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,28,30,22,29,21,19)(18,24,25,23,27,32,26,31);(9,11)(10,13)(12,15)(14,16)(17,21,22,28)(18,25,27,26)(19,20,30,29)(23,24,31,32);(11,15)(13,16)(17,21,22,28)(18,25,27,26)(19,29,30,20)(23,32,31,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513215);