#8159832

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	C₂⁵	C₂⁴
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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8067028);