#9007903

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	#43298	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,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, 8915099);