#8158977

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

3848 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=gh,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, 8066173);