#8110602

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(18,23)(20,29)(24,30)(27,32)(33,34)(35,41)(36,38)(37,44)(39,43)(40,45)(42,46)(47,48);(2,8)(3,4)(10,13)(11,12)(19,26)(24,30)(27,32)(28,31)(34,36)(39,41)(40,44)(46,47);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,18,22,23)(19,24,26,30)(20,25,29,21)(27,31,32,28)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,19)(18,24)(20,27)(21,28)(22,26)(23,30)(25,31)(29,32)(33,34,38,36)(35,39,43,41)(37,40,45,44)(42,46,48,47);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,20)(18,25)(19,27)(21,23)(22,29)(24,31)(26,32)(28,30)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,21,22,25)(18,20,23,29)(19,28,26,31)(24,27,30,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017798);