#10188275

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#53704	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10095471);