#8551539

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	#52316	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=hi,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, 8458735);