#8395008

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

4126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8302204);