#8535133

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	#36987	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,e^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, 8442329);