#455709

order	512
nilpotency class	3
isoclinism	isoclinic class of #454041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1069 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362905);