#8118931

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

3940 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026127);