#8602619

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8509815);