#8617758

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	#56769	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8524954);