#8772708

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,e^2=h,f^2=h,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, 8679904);