#8105631

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

3983 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)(4,9)(5,8)(6,10)(7,14)(12,15)(13,16)(17,18,21,23)(19,25,26,22)(20,28,29,24)(27,30,32,31)(33,34,38,42)(35,39,40,36)(37,41,47,48)(43,46,45,44);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(17,18)(19,22)(20,28)(21,23)(24,29)(25,26)(27,31)(30,32)(33,35)(34,39)(36,42)(37,45)(38,40)(41,44)(43,47)(46,48);(1,2)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(17,19,21,26)(18,22,23,25)(20,27,29,32)(24,30,28,31)(33,36,38,39)(34,40,42,35)(37,46,47,44)(41,43,48,45);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(18,23)(19,26)(24,28)(27,32)(33,37,38,47)(34,41,42,48)(35,43,40,45)(36,44,39,46);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,11,16)(17,20,21,29)(18,24,23,28)(19,27,26,32)(22,30,25,31)(34,42)(35,40)(41,48)(43,45);(1,5,6,4)(2,8,10,9)(3,13,14,12)(7,15,11,16)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^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, 8012827);