#512791

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₃₂	C₂³
commutator, socle	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#121	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 72 points:

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

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, 419987);