#8755318

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	#56018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=ghi,e^2=gh,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, 8662514);