#8125479

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	#47700	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8032675);