#127084

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
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	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	C₂×#1282
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280935, #72280958, #72280974, …
maximal normal subgroup of	#72280935, #72280958, #72280974, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34280);