#8750726

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	#53080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,e^2=g,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, 8657922);