#8105509

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	#52277	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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)(3,11)(4,5)(6,9)(7,10)(8,15)(12,14)(13,16)(17,18,23,26)(19,22,25,21)(20,28,30,24)(27,31,32,29)(33,34,37,40)(35,38,42,47)(36,43,45,39)(41,48,44,46);(1,3,7,15)(2,8,10,11)(4,12,9,16)(5,13,6,14)(17,19,23,25)(18,21,26,22)(20,27,30,32)(24,31,28,29)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4,7,9)(2,5,10,6)(3,12,15,16)(8,13,11,14)(17,20,23,30)(18,24,26,28)(19,27,25,32)(21,31,22,29)(33,36)(34,39)(35,41)(37,45)(38,46)(40,43)(42,44)(47,48);(1,5,7,6)(2,9,10,4)(3,13,15,14)(8,16,11,12)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,28,32)(35,42)(36,45)(38,47)(39,43);(1,5,7,6)(2,4,10,9)(3,14,15,13)(8,16,11,12)(17,22,23,21)(18,19,26,25)(20,29,30,31)(24,27,28,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,9,10,4)(3,14,15,13)(8,16,11,12)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,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, 8012705);