#123466

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#385
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,4)(9,10,13,19,15,18,14,20)(11,21,24,34,23,16,28,30)(12,17,26,31,25,33,27,32)(22,29,37,39,36,35,38,40);(9,11,14,24,15,23,13,28)(10,16,20,30,18,21,19,34)(12,22,27,37,25,36,26,38)(17,29,32,39,33,35,31,40);(2,4)(5,6,7,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,25)(16,29)(18,33)(19,31)(20,32)(21,35)(23,36)(24,37)(28,38)(30,39)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30662);