#8521595

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	#55732	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8428791);