#8157365

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064561);