#6402140

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	#2543	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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)(9,10,13,16)(11,15,12,14)(17,18,23,21,22,24,20,26)(19,27,28,29,30,25,31,32);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,15)(13,16)(18,24)(21,26)(25,27)(29,32);(2,6)(5,8)(9,11,13,12)(10,14,16,15)(17,19,22,30)(18,25,24,27)(20,31,23,28)(21,32,26,29);(9,11,13,12)(10,15,16,14)(17,20,22,23)(18,26,24,21)(19,28,30,31)(25,32,27,29);(9,12,13,11)(10,15,16,14)(17,21,23,24,22,26,20,18)(19,29,28,25,30,32,31,27)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6309336);