#6406515

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	#2676	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 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)(11,16)(12,14)(13,15)(17,18,23,28,22,27,20,26)(19,25,29,21,30,24,31,32);(2,5)(3,7)(17,19)(18,24)(20,31)(21,26)(22,30)(23,29)(25,27)(28,32);(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,31)(21,28)(22,30)(23,29)(24,27)(26,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,20,22,23)(18,26,27,28)(19,29,30,31)(21,24,32,25);(2,5)(6,8)(17,21,23,24,22,32,20,25)(18,19,28,29,27,30,26,31)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^b=ei,d^c=di,e^d=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, 6313711);