#8165355

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	#59675	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4012 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

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, 8072551);