#8155044

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

4257 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8062240);