#8105642

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(8,14)(12,15)(13,16)(17,18,23,26)(19,22,25,21)(20,28,30,24)(27,31,32,29)(33,34)(35,42)(36,40)(37,46)(38,41)(39,44)(43,48)(45,47);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,16)(10,11)(17,19,23,25)(18,21,26,22)(20,27,30,32)(24,31,28,29)(33,35)(34,39)(36,45)(37,43)(38,44)(40,48)(41,42)(46,47);(1,4,7,9)(2,5,10,6)(3,12,14,16)(8,13,11,15)(17,20)(18,24)(19,27)(21,31)(22,29)(23,30)(25,32)(26,28)(33,36,38,37)(34,40,41,46)(35,43,44,45)(39,47,42,48);(1,5,7,6)(2,9,10,4)(3,13,14,15)(8,16,11,12)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37,38,36)(34,40,41,46)(35,43,44,45)(39,48,42,47);(3,14)(8,11)(12,16)(13,15)(17,22,23,21)(18,19,26,25)(20,29,30,31)(24,27,28,32);(1,6,7,5)(2,9,10,4)(3,15,14,13)(8,16,11,12)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=h,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, 8012838);