#8493056

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

4260 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)(3,9,4,14)(5,6,11,10)(12,13,15,16)(18,24)(19,20)(26,29)(27,28)(33,34,39,43)(35,40,44,48)(36,38,42,47)(37,46,45,41);(2,8)(5,11)(9,14)(12,15)(17,18)(19,25)(20,22)(21,29)(23,24)(26,31)(27,30)(28,32)(35,44)(37,45)(40,48)(41,46);(1,3,7,4)(2,9,8,14)(5,12,11,15)(6,13,10,16)(17,19,23,20)(18,25,24,22)(21,27,31,28)(26,32,29,30)(33,35,39,44)(34,40,43,48)(36,45,42,37)(38,41,47,46);(1,4,7,3)(2,9,8,14)(5,15,11,12)(6,13,10,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,36)(34,38)(35,45)(37,44)(39,42)(40,41)(43,47)(46,48);(1,5,7,11)(2,10,8,6)(3,12,4,15)(9,16,14,13)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,37,39,45)(34,41,43,46)(35,36,44,42)(38,40,47,48);(1,6,7,10)(2,11,8,5)(3,13,4,16)(9,15,14,12)(17,22,23,25)(18,20,24,19)(21,30,31,32)(26,28,29,27)(33,38)(34,42)(35,41)(36,43)(37,40)(39,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8400252);