#8137581

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	#57050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(18,23)(19,28)(20,30)(21,31)(37,45)(40,47)(43,46)(44,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,18,22,23)(19,24,28,25)(20,26,30,32)(21,27,31,29)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(17,19)(18,24)(20,29)(21,26)(22,28)(23,25)(27,30)(31,32)(33,35,38,39)(34,36,41,42)(37,43,45,46)(40,44,47,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,19)(18,25)(20,27)(21,26)(22,28)(23,24)(29,30)(31,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20)(18,26)(19,29)(21,24)(22,30)(23,32)(25,31)(27,28)(33,36)(34,39)(35,41)(37,44)(38,42)(40,46)(43,47)(45,48);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,21,22,31)(18,27,23,29)(19,26,28,32)(20,24,30,25)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044777);