#8135671

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

4234 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)(3,10)(4,8)(5,15)(6,16)(7,9)(11,12)(13,14)(20,28)(24,31)(26,32)(29,30)(33,34)(35,44)(36,41)(37,47)(38,48)(39,43)(40,46)(42,45);(3,4)(8,10)(11,13)(12,14)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,35,39,46)(34,40,43,44)(36,45,47,38)(37,42,41,48);(1,3,7,4)(2,8,9,10)(5,11,16,13)(6,12,15,14)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,36,39,47)(34,41,43,37)(35,45,46,38)(40,48,44,42);(1,4,7,3)(2,8,9,10)(5,11,16,13)(6,14,15,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37)(34,36)(35,42)(38,44)(39,41)(40,45)(43,47)(46,48);(1,5)(2,6)(3,11)(4,13)(7,16)(8,12)(9,15)(10,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,34)(35,40)(36,41)(37,47)(38,42)(39,43)(44,46)(45,48);(1,6)(2,5)(3,12)(4,14)(7,15)(8,11)(9,16)(10,13)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,42)(35,36)(37,44)(39,45)(40,41)(43,48)(46,47)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,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, 8042867);