#8417061

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3775 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)(3,13)(4,6)(5,10)(7,11)(8,12)(9,16)(14,15)(18,22)(23,28)(24,29)(30,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,3,8,16)(2,9,12,13)(4,14,5,11)(6,7,10,15)(19,25)(23,28)(26,31)(30,32)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(1,4,8,5)(2,10,12,6)(3,14,16,11)(7,9,15,13)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,5,8,4)(2,10,12,6)(3,11,16,14)(7,9,15,13)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,6,8,10)(2,5,12,4)(3,15,16,7)(9,14,13,11)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,7,8,15)(2,11,12,14)(3,10,16,6)(4,13,5,9)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324257);