#8127573

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	#56338	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,f^2=h,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^c=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, 8034769);