#8105617

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	#54879	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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)(6,10)(7,9)(8,14)(12,13)(15,16)(17,18,23,26)(19,22,25,21)(20,28,30,24)(27,31,32,29)(33,34,37,39)(35,38,41,46)(36,40,44,47)(42,45,43,48);(1,3,7,14)(2,8,9,11)(4,12,10,16)(5,15,6,13)(17,19,23,25)(18,21,26,22)(20,27,30,32)(24,31,28,29)(33,35)(34,38)(36,43)(37,41)(39,46)(40,48)(42,44)(45,47);(1,4,7,10)(2,5,9,6)(3,12,14,16)(8,15,11,13)(17,20,23,30)(18,24,26,28)(19,27,25,32)(21,31,22,29)(34,39)(35,41)(36,44)(45,48);(2,9)(4,10)(8,11)(12,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,36,37,44)(34,40,39,47)(35,42,41,43)(38,45,46,48);(1,5,7,6)(2,4,9,10)(3,13,14,15)(8,16,11,12)(17,22,23,21)(18,19,26,25)(20,29,30,31)(24,27,28,32)(34,39)(35,41)(40,47)(42,43);(1,6,7,5)(2,10,9,4)(3,13,14,15)(8,16,11,12)

Polycyclic presentation:

2^9|a^2=h,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, 8012813);