#8932189

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839385);