#7144320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7051516);