#8105556

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

4190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^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, 8012752);