#8202281

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3853 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8109477);