#8135656

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

4184 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8042852);