#505181

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
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	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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)(3,6)(4,5)(7,8)(9,10)(11,24)(12,31)(13,38)(14,22)(15,21)(16,20)(17,23)(18,27)(19,35)(25,43)(26,61)(28,47)(29,44)(30,45)(32,48)(33,49)(34,52)(36,65)(37,50)(39,54)(40,53)(41,56)(42,51)(46,62)(55,59)(57,71)(58,69)(60,70)(63,64)(66,72)(67,68);(2,5)(6,8)(9,11,15,28,17,30,14,27)(10,18,22,45,23,47,21,24)(12,32,35,57,37,66,34,63)(13,26,39,62,40,29,16,41)(19,48,31,64,52,72,50,71)(20,44,53,46,54,61,38,56)(25,42,59,33,60,67,58,36)(43,65,69,68,70,49,55,51);(1,3,4,7)(2,6,5,8)(9,12,14,34,17,37,15,35)(10,19,21,50,23,52,22,31)(11,25,27,58,30,60,28,59)(13,33,16,42,40,36,39,67)(18,43,24,55,47,70,45,69)(20,49,38,68,54,65,53,51)(26,57,41,32,29,63,62,66)(44,48,56,71,61,72,46,64);(2,5)(6,8)(9,13,14,16,17,40,15,39)(10,20,21,38,23,54,22,53)(11,26,27,41,30,29,28,62)(12,33,34,42,37,36,35,67)(18,44,24,56,47,61,45,46)(19,49,50,68,52,65,31,51)(25,57,58,32,60,63,59,66)(43,48,55,71,70,72,69,64)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=fi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=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, 412377);