#8105503

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

4166 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)(6,10)(7,9)(8,14)(12,13)(15,16)(17,18,22,25)(19,24,20,23)(21,29,30,26)(27,31,28,32)(33,34,37,40)(35,38,42,47)(36,43,45,39)(41,48,44,46);(1,3,7,14)(2,8,9,11)(4,12,10,16)(5,15,6,13)(17,19,22,20)(18,23,25,24)(21,28,30,27)(26,32,29,31)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4,7,10)(2,5,9,6)(3,12,14,16)(8,15,11,13)(17,20,22,19)(18,24,25,23)(21,28,30,27)(26,32,29,31)(33,36)(34,39)(35,41)(37,45)(38,46)(40,43)(42,44)(47,48);(2,9)(4,10)(8,11)(12,16)(18,25)(19,20)(26,29)(27,28)(35,42)(36,45)(38,47)(39,43);(1,5,7,6)(2,4,9,10)(3,13,14,15)(8,16,11,12)(17,21,22,30)(18,26,25,29)(19,27,20,28)(23,31,24,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,10,9,4)(3,13,14,15)(8,16,11,12)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^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, 8012699);