#8506268

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	#36873	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^2=gi,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, 8413464);