#8750753

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3815 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,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, 8657949);