#512731

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

390 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)(11,16,22,32,23,33,44,54,24,34,45,55,46,56,64,69,21,31,41,51,42,52,61,66,43,53,62,67,63,68,71,72);(2,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,31)(18,32)(19,33)(20,34)(25,44)(26,45)(27,41)(28,46)(29,42)(30,43)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(47,64)(48,61)(49,62)(50,63)(57,66)(58,67)(59,68)(60,69)(65,71)(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);(11,21)(16,31)(22,41)(23,42)(24,43)(32,51)(33,52)(34,53)(44,61)(45,62)(46,63)(54,66)(55,67)(56,68)(64,71)(69,72)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231857, #72231859, #72231861, …
maximal normal subgroup of	#72231857, #72231859, #72231861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419927);