#8104937

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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)(3,11,14,8)(5,10,6,9)(12,15,13,16)(17,18,23,26)(19,22,25,21)(20,28,30,24)(27,31,32,29);(1,3,7,14)(2,8,4,11)(5,12,6,13)(9,15,10,16)(17,19,23,25)(18,21,26,22)(20,27,30,32)(24,31,28,29)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,7,2)(3,11,14,8)(5,9,6,10)(12,15,13,16)(17,20,23,30)(18,24,26,28)(19,27,25,32)(21,31,22,29)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(1,5,7,6)(2,9,4,10)(3,12,14,13)(8,15,11,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,5,7,6)(2,10,4,9)(3,13,14,12)(8,15,11,16)(17,22,23,21)(18,19,26,25)(20,29,30,31)(24,27,28,32)(34,40)(38,41)(39,48)(46,47);(1,6,7,5)(2,9,4,10)(3,13,14,12)(8,15,11,16)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8012133);