#8757371

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	#56400	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(17,18)(19,23)(20,26)(21,24)(22,30)(25,28)(27,29)(31,32)(34,37)(39,42)(40,45)(46,48);(1,3)(2,8)(4,14)(5,12)(6,15)(7,13)(9,16)(10,11)(17,19)(18,25)(20,27)(21,23)(22,31)(24,28)(26,32)(29,30)(36,44)(40,45)(41,47)(46,48);(2,9)(3,13)(4,12)(11,15)(17,20,24,22)(18,26,21,30)(19,27,28,31)(23,29,25,32)(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)(17,21,24,18)(19,23,28,25)(20,26,22,30)(27,32,31,29)(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)(17,22,24,20)(18,26,21,30)(19,27,28,31)(23,32,25,29)(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)(17,23)(18,19)(20,29)(21,28)(22,32)(24,25)(26,27)(30,31)(33,37,38,34)(35,42,43,39)(36,45,44,40)(41,48,47,46)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gh,e^2=gi,f^2=h,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, 8664567);