#8898646

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	#47911	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=g,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, 8805842);