#398613

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305809);