#8908056

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	#50091	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815252);