#8459029

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	#40129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8366225);