#8159075

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	#58855	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=g,e^2=gh,f^2=h,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, 8066271);