#8757375

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	#47310	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664571);