#6402131

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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

1425 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)(17,18,24,27,23,22,21,32)(19,28,29,31,30,25,20,26);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,15)(13,16)(17,19,23,30)(18,25,22,28)(20,24,29,21)(26,32,31,27);(2,6)(5,8)(9,11,13,12)(10,14,16,15)(17,20,23,29)(18,26,22,31)(19,21,30,24)(25,27,28,32);(9,11,13,12)(10,15,16,14)(17,21,23,24)(18,27,22,32)(19,29,30,20)(25,31,28,26);(9,12,13,11)(10,15,16,14)(17,22,24,32,23,18,21,27)(19,25,29,26,30,28,20,31)

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,d^a=di,e^a=efi,c^b=ci,d^b=df,e^b=ei,d^c=di,e^d=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, 6309327);