#8417314

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	#42416	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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)(3,10)(4,5)(6,7)(8,13)(9,15)(11,12)(14,16)(19,27)(20,29)(23,30)(24,32)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,45)(46,47);(1,2)(3,8)(4,5)(6,7)(9,15)(10,13)(11,16)(12,14)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,28,31)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(2,7)(5,9)(8,10)(12,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,32)(26,30)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,3,6,13)(2,8,7,10)(4,14,15,11)(5,12,9,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,4)(2,9)(3,11)(5,7)(6,15)(8,12)(10,16)(13,14)(17,20,22,29)(18,24,26,32)(19,28,27,21)(23,31,30,25)(33,35,39,38)(34,42,43,40)(36,44,41,46)(37,45,48,47);(1,5,6,9)(2,4,7,15)(3,12,13,16)(8,14,10,11)(17,21,22,28)(18,25,26,31)(19,29,27,20)(23,32,30,24)(33,38,39,35)(34,40,43,42)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324510);