#8898924

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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:

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

Polycyclic presentation:

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

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, 8806120);