#8100333

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,10,12,8)(5,14,6,9)(11,15,13,16)(17,18,22,25)(19,21,27,23)(20,28,30,24)(26,29,32,31)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3,7,12)(2,8,4,10)(5,11,6,13)(9,15,14,16)(17,19,22,27)(18,23,25,21)(20,26,30,32)(24,29,28,31)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(1,4,7,2)(3,10,12,8)(5,9,6,14)(11,15,13,16)(17,20,22,30)(18,24,25,28)(19,26,27,32)(21,31,23,29)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47);(1,5,7,6)(2,9,4,14)(3,11,12,13)(8,15,10,16)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,32,28,26)(36,44)(39,48)(41,43)(45,47);(3,12)(8,10)(11,13)(15,16)(18,25)(19,27)(24,28)(26,32)(35,42)(38,46)(41,43)(45,47);(1,6,7,5)(2,9,4,14)(3,13,12,11)(8,15,10,16)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^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, 8007529);