#8105498

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

4155 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 8012694);