#8696759

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	#60033	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4121 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603955);