#6402598

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,24,22,25,20,29)(19,27,26,31,28,21,30,32);(1,3,4,7)(2,5,6,8)(9,10,13,14)(11,15,12,16)(17,19,22,28)(18,21,25,27)(20,30,23,26)(24,32,29,31);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,20,22,23)(18,24,25,29)(19,26,28,30)(21,31,27,32);(10,14)(15,16)(18,25)(19,28)(24,29)(26,30);(9,12,13,11)(10,16,14,15)(17,21,20,31,22,27,23,32)(18,26,29,19,25,30,24,28)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df

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, 6309794);