#8720805

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	#56439	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8628001);