#8504733

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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)(3,9)(4,11)(5,8)(6,7)(10,16)(12,13)(14,15)(17,18,23,21)(19,26,22,24)(20,29,30,25)(27,31,28,32)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(2,7)(4,9)(8,15)(12,16)(17,19,23,22)(18,24,21,26)(20,27,30,28)(25,31,29,32)(34,39)(35,42)(41,44)(43,48);(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3)(2,4)(5,13)(6,11)(7,9)(8,16)(10,14)(12,15)(17,20)(18,25)(19,28)(21,29)(22,27)(23,30)(24,32)(26,31)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21,23,18)(19,24,22,26)(20,29,30,25)(27,31,28,32);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22,23,19)(18,24,21,26)(20,28,30,27)(25,31,29,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8411929);