#8824363

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3831 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731559);