#8184783

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	#53060	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4135 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)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(18,20)(19,25)(24,28)(27,31)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(2,7)(5,8)(9,12)(13,15)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8091979);