#8105618

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

4174 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:

(1,2,7,4)(3,11,13,8)(5,9,6,10)(12,16,14,15)(18,23)(19,27)(20,21)(31,32)(33,34,37,39)(35,38,41,46)(36,40,44,47)(42,45,43,48);(1,3,7,13)(2,8,4,11)(5,14,6,12)(9,16,10,15)(17,18,22,23)(19,24,27,30)(20,26,21,25)(28,32,29,31)(33,35)(34,38)(36,43)(37,41)(39,46)(40,48)(42,44)(45,47);(1,4,7,2)(3,11,13,8)(5,10,6,9)(12,16,14,15)(17,19,22,27)(18,24,23,30)(20,29,21,28)(25,32,26,31)(34,39)(35,41)(36,44)(45,48);(2,4)(8,11)(9,10)(15,16)(17,20,22,21)(18,25,23,26)(19,28,27,29)(24,31,30,32)(33,36,37,44)(34,40,39,47)(35,42,41,43)(38,45,46,48);(1,5,7,6)(2,9,4,10)(3,12,13,14)(8,15,11,16)(17,21,22,20)(18,26,23,25)(19,28,27,29)(24,31,30,32)(34,39)(35,41)(40,47)(42,43);(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=g,c^2=g,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, 8012814);