#8112420

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	#57222	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019616);