#94119

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	#346
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 80 points:

(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,27,22,28,40,53,23,29,41,54,43,56,68,76,24,30,42,55,44,57,69,77,20,26,37,50,38,51,65,74)(19,31,33,58,34,59,62,78,32,48,60,52,61,72,79,75,35,49,39,66,63,73,67,80,36,25,45,46,64,47,70,71);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,25)(20,36)(21,39)(22,34)(23,32)(24,35)(26,49)(27,52)(28,47)(29,31)(30,48)(33,42)(37,60)(38,64)(40,67)(41,45)(43,61)(44,63)(46,55)(50,58)(51,73)(53,75)(54,66)(56,59)(57,72)(62,69)(65,79)(68,70)(71,77)(74,78)(76,80)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=cf,c^b=ci,d^b=dfi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1315);