#8322119

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	#45701	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(33,34)(35,41)(36,38)(37,40)(39,43)(42,46)(44,48)(45,47);(2,7)(3,12)(9,14)(10,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,35,39,37)(34,40,43,41)(36,44,46,47)(38,45,42,48);(2,7)(5,16)(8,11)(9,14)(18,23)(21,31)(24,29)(25,32)(33,35,39,37)(34,41,43,40)(36,44,46,47)(38,48,42,45);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19)(18,24)(20,26)(21,25)(22,28)(23,29)(27,30)(31,32)(33,36)(34,42)(35,44)(37,47)(38,43)(39,46)(40,48)(41,45);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20)(18,25)(19,27)(21,29)(22,30)(23,32)(24,31)(26,28)(33,37,39,35)(34,40,43,41)(36,44,46,47)(38,48,42,45);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,29,30,24)(33,38,39,42)(34,36,43,46)(35,45,37,48)(40,44,41,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8229315);