#8393280

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	#59676	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gi,e^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, 8300476);