#8810097

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

3899 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717293);