#509658

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3155	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

571 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:

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

Polycyclic presentation:

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

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, 416854);