#8105031

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

3990 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)(8,15)(12,14)(13,16)(18,23)(19,27)(20,21)(31,32);(1,3,7,15)(2,8,10,11)(4,12,9,16)(5,13,6,14)(17,18,22,23)(19,24,27,30)(20,26,21,25)(28,32,29,31)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,4,7,9)(2,5,10,6)(3,12,15,16)(8,13,11,14)(17,19,22,27)(18,24,23,30)(20,29,21,28)(25,32,26,31)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(1,5,7,6)(2,9,10,4)(3,13,15,14)(8,16,11,12)(17,20,22,21)(18,25,23,26)(19,28,27,29)(24,31,30,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,5,7,6)(2,4,10,9)(3,14,15,13)(8,16,11,12)(17,21,22,20)(18,26,23,25)(19,28,27,29)(24,31,30,32)(34,40)(38,41)(39,48)(46,47);(1,6,7,5)(2,9,10,4)(3,14,15,13)(8,16,11,12)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=gi,e^2=g,f^2=gh,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, 8012227);