#504665

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

510 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)(3,8)(4,6)(5,7)(9,10)(11,23)(12,28)(13,33)(14,36)(15,20)(16,21)(17,22)(18,26)(19,31)(24,54)(25,53)(27,44)(29,62)(30,61)(32,49)(34,51)(35,66)(37,43)(38,48)(39,45)(40,50)(41,58)(42,59)(46,64)(47,65)(52,56)(55,72)(57,70)(60,63)(67,69)(68,71);(1,3,5,4)(2,6,7,8)(9,11,14,26,17,27,16,39)(10,18,21,23,22,45,36,44)(12,29,31,63,32,64,40,55)(13,25,15,37,35,59,34,67)(19,46,28,60,50,62,49,72)(20,42,33,43,51,53,66,69)(24,38,56,65,58,68,57,30)(41,61,52,71,54,47,70,48);(1,4,5,3)(2,6,7,8)(9,12,14,31,17,32,16,40)(10,19,21,28,22,50,36,49)(11,24,26,56,27,58,39,57)(13,30,15,38,35,65,34,68)(18,41,23,52,45,54,44,70)(20,47,33,48,51,61,66,71)(25,55,37,29,59,63,67,64)(42,46,43,60,53,62,69,72);(2,7)(6,8)(9,13,16,34,17,35,14,15)(10,20,36,66,22,51,21,33)(11,25,39,67,27,59,26,37)(12,30,40,68,32,65,31,38)(18,42,44,69,45,53,23,43)(19,47,49,71,50,61,28,48)(24,55,57,64,58,63,56,29)(41,46,70,72,54,62,52,60)

Polycyclic presentation:

2^9|b^2=e,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, 411861);