#8105634

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,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, 8012830);