#6571619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533141
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6478815);