#8534930

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3805 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)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,2,7,5)(3,9,4,8)(6,10,14,13)(11,16,12,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,29)(30,32)(34,35)(39,41)(40,42)(46,47);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,31)(30,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,29,26,31);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8442126);