#6405516

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2703	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1396 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)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,23,21,22,25,32,31)(19,24,27,26,20,28,30,29);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(17,19,22,20)(18,24,25,28)(21,26,31,29)(23,27,32,30);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,19)(18,24,25,28)(21,26,31,29)(23,30,32,27);(9,11,12,15)(10,14,13,16)(19,20)(24,28)(26,29)(27,30);(10,13)(14,16)(17,21,23,25,22,31,32,18)(19,26,27,28,20,29,30,24)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi,d^c=di,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6312712);