#8811175

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	#48707	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 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)(4,10)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18,23,27)(19,22,28,24)(20,29,31,25)(21,26,32,30)(33,34,39,36)(35,42,41,38)(37,40,45,44)(43,48,47,46);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(17,19,23,28)(18,24,27,22)(20,21,31,32)(25,26,29,30)(37,45)(40,44)(43,47)(46,48);(2,9)(3,4)(5,11)(13,15)(17,20,23,31)(18,25,27,29)(19,21,28,32)(22,30,24,26)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,20)(18,25)(19,21)(22,30)(23,31)(24,26)(27,29)(28,32)(33,36,39,34)(35,42,41,38)(37,44,45,40)(43,48,47,46);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,21)(18,26)(19,20)(22,29)(23,32)(24,25)(27,30)(28,31)(33,37,39,45)(34,40,36,44)(35,43,41,47)(38,46,42,48);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,22,23,24)(18,19,27,28)(20,30,31,26)(21,29,32,25)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8718371);