#8749665

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3881 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)(3,10,6,8)(4,5,14,15)(11,12,13,16)(17,18)(19,21)(20,25)(22,26)(23,27)(24,30)(28,31)(29,32)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,3,7,6)(2,8,9,10)(4,13,14,11)(5,12,15,16)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,26)(31,32)(34,38)(35,42)(40,44)(43,48);(2,9)(4,14)(8,10)(11,13)(17,20,23,29)(18,25,27,32)(19,28,30,22)(21,31,24,26)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,4)(2,5)(3,11)(6,13)(7,14)(8,16)(9,15)(10,12)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,26)(27,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,5)(2,4)(3,12)(6,16)(7,15)(8,13)(9,14)(10,11)(17,21,23,24)(18,19,27,30)(20,31,29,26)(22,25,28,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,6,7,3)(2,8,9,10)(4,11,14,13)(5,12,15,16)(17,22,23,28)(18,26,27,31)(19,29,30,20)(21,32,24,25)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8656861);