#7569287

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(11,12)(14,16)(17,18)(19,24)(20,32)(21,36)(22,26)(23,27)(25,34)(28,42)(29,40)(30,39)(31,41)(33,38)(35,43)(37,44)(45,47)(46,48);(2,5)(6,8)(9,10,13,15)(11,16,12,14)(17,19,22,31)(18,24,26,41)(20,33,34,28)(21,30,23,29)(25,42,32,38)(27,40,36,39)(35,45,37,46)(43,47,44,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,15,16)(17,20)(18,25)(19,28)(21,37)(22,34)(23,35)(24,38)(26,32)(27,44)(29,46)(30,45)(31,33)(36,43)(39,48)(40,47)(41,42);(9,12,13,11)(10,14,15,16)(18,26)(19,29)(20,34)(21,23)(24,39)(28,45)(30,31)(33,46)(38,47)(40,41)(42,48)(43,44);(2,5)(6,8)(17,21,22,23)(18,27,26,36)(19,30,31,29)(20,35,34,37)(24,40,41,39)(25,43,32,44)(28,46,33,45)(38,48,42,47)

Polycyclic presentation:

2^9|b^2=g,c^2=fgi,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7476483);