#8322114

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(33,34,40,44)(35,43,38,39)(36,42,37,41)(45,48,46,47);(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)(33,35,40,38)(34,39,44,43)(36,45,37,46)(41,48,42,47);(4,12)(7,14)(10,16)(13,15)(18,23)(21,31)(24,29)(25,32)(33,36,40,37)(34,41,44,42)(35,45,38,46)(39,48,43,47);(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)(33,37,40,36)(34,42,44,41)(35,45,38,46)(39,48,43,47);(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)(33,38,40,35)(34,43,44,39)(36,46,37,45)(41,47,42,48);(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)(33,39,40,43)(34,35,44,38)(36,47,37,48)(41,46,42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,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, 8229310);