#8339939

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

3837 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8247135);