#8258577

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	#35133	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,8)(4,10)(5,12)(6,11)(7,9)(13,16)(14,15)(17,18,22,27)(19,26,29,23)(20,25,21,24)(28,31,30,32)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(1,3)(2,8)(4,6)(5,9)(7,12)(10,11)(13,14)(15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,32)(26,27)(34,36)(35,41)(40,44)(43,47);(2,9)(3,12)(6,13)(10,15)(17,20,22,21)(18,24,27,25)(19,28,29,30)(23,31,26,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4)(2,10)(3,13)(5,11)(6,12)(7,14)(8,16)(9,15)(17,21,22,20)(18,25,27,24)(19,28,29,30)(23,31,26,32)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,5)(2,3)(4,11)(6,15)(7,8)(9,12)(10,13)(14,16)(17,18)(19,26)(20,24)(21,25)(22,27)(23,29)(28,32)(30,31)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,6,7,13)(2,11,9,16)(3,4,12,14)(5,15,8,10)(17,19)(18,26)(20,30)(21,28)(22,29)(23,27)(24,31)(25,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8165773);