#8898917

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	#35421	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,21)(20,25)(22,31)(23,24)(26,30)(27,28)(29,32)(33,34,40,44)(35,45,48,41)(36,42,37,39)(38,43,47,46);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,13,16,15)(17,19,23,22)(18,21,24,31)(20,26,29,28)(25,30,32,27)(33,35)(34,41)(36,43)(37,46)(38,42)(39,47)(40,48)(44,45);(2,9)(6,16)(8,10)(13,15)(18,24)(19,22)(25,32)(26,28)(33,36,40,37)(34,42,44,39)(35,43,48,46)(38,45,47,41);(1,4,7,3)(2,10,9,8)(5,12,11,14)(6,15,16,13)(17,20)(18,25)(19,26)(21,30)(22,28)(23,29)(24,32)(27,31)(33,37,40,36)(34,42,44,39)(35,46,48,43)(38,45,47,41);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,15,10,13)(17,21)(18,19)(20,27)(22,24)(23,31)(25,28)(26,32)(29,30)(33,38,40,47)(34,43,44,46)(35,42,48,39)(36,45,37,41);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,12)(10,14)(17,22,23,19)(18,21,24,31)(20,28,29,26)(25,30,32,27)(33,39,40,42)(34,36,44,37)(35,47,48,38)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806113);