#8720804

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(18,23)(19,28)(20,27)(30,31)(33,34,40,37)(35,41,45,38)(36,39,42,47)(43,48,46,44);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,13)(15,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,31)(30,32)(33,35,40,45)(34,41,37,38)(36,46,42,43)(39,44,47,48);(2,7)(3,8)(9,16)(11,13)(17,19,22,28)(18,24,23,25)(20,29,27,32)(21,30,26,31)(33,36,40,42)(34,39,37,47)(35,43,45,46)(38,44,41,48);(1,3,6,8)(2,4,7,10)(5,13,15,11)(9,14,16,12)(17,19)(18,25)(20,29)(21,31)(22,28)(23,24)(26,30)(27,32)(33,37,40,34)(35,38,45,41)(36,39,42,47)(43,48,46,44);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,20)(18,26)(19,29)(21,23)(22,27)(24,31)(25,30)(28,32)(33,38)(34,35)(36,44)(37,45)(39,43)(40,41)(42,48)(46,47);(1,5,6,15)(2,9,7,16)(3,11,8,13)(4,12,10,14)(17,21,22,26)(18,27,23,20)(19,30,28,31)(24,32,25,29)(33,39)(34,42)(35,44)(36,37)(38,46)(40,47)(41,43)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,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, 8628000);