#492970

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×C₈×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1065 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,h^2=i,b^a=be,g^a=gi,e^b=ei,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319724, #72319733, #72319748, …
maximal normal subgroup of	#72319724, #72319733, #72319748, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400166);