#8810198

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

3839 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=gh,e^2=h,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, 8717394);