#8754998

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18)(19,20)(21,29)(22,23)(24,27)(25,30)(26,31)(28,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,7)(5,8)(9,12)(13,15)(18,23)(20,24)(25,29)(28,31)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(2,7)(3,10)(9,12)(11,16)(17,18,22,23)(19,24,27,20)(21,25,30,29)(26,31,32,28)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,29)(30,32);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662194);