#8418453

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	#42503	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3936 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)(3,11)(4,9)(5,10)(6,8)(7,14)(12,16)(13,15)(17,18,22,25)(19,21,24,26)(20,23,31,29)(27,30,28,32)(33,34,39,41)(35,40,36,44)(37,38,47,48)(42,46,45,43);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,13,16,9)(17,19)(18,21)(20,28)(22,24)(23,32)(25,26)(27,31)(29,30)(33,35,39,36)(34,40,41,44)(37,45,47,42)(38,43,48,46);(2,8)(5,16)(7,11)(9,13)(19,24)(21,26)(27,28)(30,32)(33,36,39,35)(34,40,41,44)(37,45,47,42)(38,46,48,43);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,11,16,7)(17,18)(19,26)(20,29)(21,24)(22,25)(23,31)(27,30)(28,32)(33,37)(34,38)(35,42)(36,45)(39,47)(40,46)(41,48)(43,44);(1,3)(2,7)(4,10)(5,13)(6,14)(8,11)(9,16)(12,15)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,29)(26,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,45)(41,47)(42,44);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,11,15,7)(17,21,22,26)(18,24,25,19)(20,30,31,32)(23,28,29,27)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,f^2=gi,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, 8325649);