#8105597

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

4155 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)(19,25)(20,28)(21,23)(22,26)(24,30)(27,31)(29,32)(33,34,37,40)(35,38,42,47)(36,43,45,39)(41,48,44,46);(1,3,7,13)(2,8,4,11)(5,14,6,12)(9,16,10,15)(17,19)(18,22)(20,29)(21,26)(23,25)(24,32)(27,30)(28,31)(33,35,37,42)(34,38,40,47)(36,44,45,41)(39,48,43,46);(1,4,7,2)(3,11,13,8)(5,10,6,9)(12,16,14,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,31)(28,32)(29,30)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(2,4)(8,11)(9,10)(15,16)(18,23)(19,26)(24,28)(27,29)(35,42)(36,45)(38,47)(39,43);(1,5,7,6)(2,9,4,10)(3,12,13,14)(8,15,11,16)(17,20,21,30)(18,24,23,28)(19,27,26,29)(22,31,25,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,10,4,9)(3,12,13,14)(8,15,11,16)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^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, 8012793);