#8898625

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(3,12)(4,15)(5,16)(6,10)(7,8)(9,14)(11,13)(17,18,22,25)(19,23,29,26)(20,21,24,30)(27,28,31,32)(35,41)(38,45)(42,47)(46,48);(1,3,8,14)(2,9,7,12)(4,13,6,5)(10,11,15,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,4,8,6)(2,10,7,15)(3,13,14,5)(9,11,12,16)(17,18,22,25)(19,26,29,23)(20,21,24,30)(27,32,31,28)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,5)(2,11)(3,4)(6,14)(7,16)(8,13)(9,15)(10,12)(17,20)(18,21)(19,27)(22,24)(23,28)(25,30)(26,32)(29,31);(1,6,8,4)(2,10,7,15)(3,13,14,5)(9,16,12,11)(17,21)(18,24)(19,28)(20,25)(22,30)(23,31)(26,27)(29,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,7,8,2)(3,12,14,9)(4,10,6,15)(5,16,13,11)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,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, 8805821);