#8534920

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	#57143	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3968 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442116);