#10195785

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#47671	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3899 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10102981);