#8754988

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3769 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:

(1,2,5,6)(3,9,10,7)(4,8,12,13)(11,15,16,14)(18,21)(19,25)(24,28)(27,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,6)(4,12)(7,9)(11,16)(17,18,22,21)(19,26,25,23)(20,28,29,24)(27,30,31,32)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(3,10)(7,9)(11,16)(14,15)(17,19,22,25)(18,23,21,26)(20,27,29,31)(24,30,28,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18,22,21)(19,23,25,26)(20,24,29,28)(27,30,31,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,29)(18,24,21,28)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,21,22,18)(19,26,25,23)(20,28,29,24)(27,32,31,30)(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=h,c^2=i,d^2=i,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, 8662184);