#127082

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	C₂×#1282
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

601 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,11,7,14,27,38,8,13,6,12,26,37,28,39)(3,15,18,41,20,43,48,62,16,40,19,42,45,61,49,63)(4,21,22,36,23,50,51,60,24,10,25,33,52,34,53,58)(9,29,31,54,32,55,57,17,30,46,35,47,56,64,59,44);(1,3,6,19,7,20,28,49,8,16,5,18,26,45,27,48)(2,9,12,31,14,32,39,57,13,30,11,35,37,56,38,59)(4,17,22,46,23,47,51,64,24,44,25,29,52,54,53,55)(10,15,33,42,34,43,58,63,21,40,36,41,50,61,60,62);(3,16)(9,30)(15,40)(17,44)(18,19)(20,45)(29,46)(31,35)(32,56)(41,42)(43,61)(47,54)(48,49)(55,64)(57,59)(62,63)

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=h,f^2=h,h^2=i,b^a=bd,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280848, #72280870, #72281023, …
maximal normal subgroup of	#72280848, #72280870, #72281023, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34278);