#8322399

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	#39539	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,e^2=gh,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, 8229595);