#8105277

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,31)(28,30)(29,32)(33,34,38,42)(35,39,40,36)(37,41,47,48)(43,46,45,44);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,29)(18,23,26,32)(20,27,21,28)(24,30,25,31)(33,35)(34,39)(36,42)(37,45)(38,40)(41,44)(43,47)(46,48);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20,22,21)(18,24,26,25)(19,27,29,28)(23,30,32,31)(33,36,38,39)(34,40,42,35)(37,46,47,44)(41,43,48,45);(2,7)(6,9)(8,14)(13,15)(17,20,22,21)(18,25,26,24)(19,28,29,27)(23,30,32,31)(33,37,38,47)(34,41,42,48)(35,43,40,45)(36,44,39,46);(3,10)(6,9)(11,16)(13,15)(17,21,22,20)(18,25,26,24)(19,28,29,27)(23,31,32,30)(34,42)(35,40)(41,48)(43,45);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=hi,f^2=g,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, 8012473);