#151886

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₁₆	C₂³
commutator	C₂²	C₂×C₄×C₁₆
socle	C₂³	C₂²×C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

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

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,c^a=ci

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59082);