#8393683

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	#49842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300879);