#8772536

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	#43373	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(3,10,9,5)(4,8,12,14)(11,16,15,13)(17,18)(19,20)(21,29)(22,23)(24,25)(26,30)(27,31)(28,32)(33,34,40,44)(35,42,37,38)(36,41,39,43)(45,46,47,48);(2,7)(4,12)(5,10)(11,15)(18,23)(19,25)(26,29)(27,32)(33,35,40,37)(34,38,44,42)(36,45,39,47)(41,48,43,46);(2,7)(3,9)(4,12)(13,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,36,40,39)(34,41,44,43)(35,45,37,47)(38,48,42,46);(1,3,6,9)(2,5,7,10)(4,11,12,15)(8,13,14,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,37,40,35)(34,42,44,38)(36,45,39,47)(41,48,43,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,38,40,42)(34,35,44,37)(36,46,39,48)(41,47,43,45);(1,5)(2,9)(3,7)(4,13)(6,10)(8,15)(11,14)(12,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,39,40,36)(34,43,44,41)(35,45,37,47)(38,48,42,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8679732);