#8617764

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	#49605	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(3,6,10,9)(4,7,13,12)(11,14,16,15)(17,18,23,27)(19,24,21,25)(20,22,26,32)(28,29,30,31)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(1,2)(3,9)(4,7)(5,8)(6,10)(11,15)(12,13)(14,16)(17,19)(18,24)(20,28)(21,23)(22,29)(25,27)(26,30)(31,32)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(3,10)(6,9)(11,16)(14,15)(19,21)(24,25)(28,30)(29,31)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,15)(14,16)(17,20)(18,22)(19,28)(21,30)(23,26)(24,29)(25,31)(27,32)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,21,23,19)(18,25,27,24)(20,30,26,28)(22,31,32,29)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,22)(18,26)(19,29)(20,27)(21,31)(23,32)(24,30)(25,28)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8524960);