#8416729

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	#55215	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=i,e^2=gi,f^2=g,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, 8323925);