#513019

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#13663	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

354 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)(11,22,23,16)(13,17,27,39)(14,33,28,60)(15,34,29,21)(19,44,42,30)(20,46,43,31)(24,48,52,35)(25,58,53,63)(26,38,54,51)(32,47,59,41)(36,55,49,68)(37,56,50,69)(40,62,45,61)(57,65,70,67)(64,71,66,72);(2,5)(6,8)(9,11,12,23)(10,16,18,22)(13,24,27,52)(14,25,28,53)(15,26,29,54)(17,35,39,48)(19,36,42,49)(20,37,43,50)(21,38,34,51)(30,55,44,68)(31,56,46,69)(32,57,59,70)(33,63,60,58)(40,64,45,66)(41,65,47,67)(61,71,62,72);(10,17)(13,30)(14,31)(15,29)(16,35)(18,39)(19,20)(21,47)(22,48)(24,55)(25,56)(26,54)(27,44)(28,46)(32,62)(33,40)(34,41)(36,37)(38,67)(42,43)(45,60)(49,50)(51,65)(52,68)(53,69)(57,72)(58,66)(59,61)(63,64)(70,71);(1,3,4,7)(2,6,5,8)(11,23)(16,22)(24,52)(25,53)(26,54)(35,48)(36,49)(37,50)(38,51)(55,68)(56,69)(57,70)(58,63)(64,66)(65,67)(71,72)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ei,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260454, #72322380
maximal normal subgroup of	#72260454, #72322380

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420215);