#8535005

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	#37493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442201);