#8105561

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4182 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)(17,18,22,23)(19,27,29,24)(20,26,21,25)(28,31,30,32)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,15)(10,16)(18,23)(20,21)(24,27)(28,30)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4,7,2)(3,11,14,8)(5,9,6,10)(12,15,13,16)(17,19)(18,24)(20,30)(21,28)(22,29)(23,27)(25,32)(26,31)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(1,5,7,6)(2,9,4,10)(3,12,14,13)(8,15,11,16)(17,20,22,21)(18,25,23,26)(19,28,29,30)(24,31,27,32)(35,42)(36,45)(38,47)(39,43);(1,5,7,6)(2,10,4,9)(3,13,14,12)(8,15,11,16)(17,21,22,20)(18,26,23,25)(19,28,29,30)(24,31,27,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,9,4,10)(3,13,14,12)(8,15,11,16)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,e^2=g,f^2=ghi,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, 8012757);