#425103

order	512
nilpotency class	3
isoclinism	isoclinic class of #417732
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332299);