#7481533

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1603 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)(17,18)(19,29)(20,33)(21,27)(22,28)(23,39)(24,32)(25,35)(26,36)(30,41)(31,43)(34,46)(37,42)(38,45)(40,47)(44,48);(2,5)(3,7)(9,10,12,13)(11,15,16,14)(17,19,22,32)(18,24,28,29)(20,34,36,30)(21,37,23,31)(25,44,45,40)(26,46,33,41)(27,43,39,42)(35,47,38,48);(9,10)(11,14)(12,13)(15,16)(17,20,22,36)(18,25,28,45)(19,30,32,34)(21,35,23,38)(24,40,29,44)(26,39,33,27)(31,47,37,48)(41,42,46,43);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20)(18,26)(19,30)(21,38)(22,36)(23,35)(24,41)(25,27)(28,33)(29,46)(31,48)(32,34)(37,47)(39,45)(40,43)(42,44);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,27,28,39)(19,31,32,37)(20,35,36,38)(24,42,29,43)(25,26,45,33)(30,47,34,48)(40,46,44,41)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7388729);