#8105311

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	#52278	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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=g,e^2=gi,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, 8012507);