#8417228

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	#54803	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(18,24)(19,21)(26,31)(27,29)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(2,3)(7,9)(8,10)(14,15)(17,18,23,24)(19,25,21,20)(22,26,30,31)(27,32,29,28)(34,39)(35,42)(41,44)(43,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,23,21)(18,25,24,20)(22,29,30,27)(26,28,31,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,31)(30,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,19)(18,25,24,20)(22,29,30,27)(26,32,31,28);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,30)(18,26,24,31)(19,27,21,29)(20,28,25,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324424);