#151895

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12599
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

311 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)(3,13,15,8)(4,10)(5,11)(6,12)(7,9)(14,28,18,25)(16,35,40,26)(17,36,41,27)(19,29)(20,32)(21,33)(22,34)(23,30)(24,31)(37,48)(38,51,42,49)(39,53,44,50)(43,57,60,52)(45,54)(46,56)(47,55)(58,62)(59,64,61,63);(1,3,4,16,5,17,20,43,6,18,21,42,22,44,46,61,7,15,19,40,23,41,45,60,24,14,37,38,47,39,58,59)(2,8,10,26,11,27,32,52,12,28,33,51,34,53,56,64,9,13,29,35,30,36,54,57,31,25,48,49,55,50,62,63);(3,14,15,18)(4,19)(8,25,13,28)(10,29)(16,42,40,38)(17,39,41,44)(20,45)(21,37)(26,51,35,49)(27,50,36,53)(32,54)(33,48)(43,61,60,59)(46,58)(52,64,57,63)(56,62)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217181, #72217183, #72217203, …
maximal normal subgroup of	#72217181, #72217183, #72217203, …
maximal quotient of	#72227023, C₂×#151895

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59091);