#8345994

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

4088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8253190);