#8202280

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109476);