#8750738

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×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3809 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657934);