#8898058

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	#58295	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(3,12)(4,15)(5,6)(7,9)(8,14)(10,13)(11,16)(33,34,40,42)(35,36,38,37)(39,47,48,41)(43,45,46,44);(1,3,7,14)(2,8,9,12)(4,11,13,5)(6,10,16,15)(17,18,22,23)(19,24,29,32)(20,25,30,28)(21,26,31,27)(33,35)(34,36)(37,42)(38,40)(39,43)(41,44)(45,47)(46,48);(2,9)(6,16)(8,12)(10,15)(18,23)(21,31)(24,32)(25,28)(33,36,40,37)(34,35,42,38)(39,45,48,44)(41,46,47,43);(1,4)(2,10)(3,11)(5,14)(6,12)(7,13)(8,16)(9,15)(17,19)(18,24)(20,26)(21,28)(22,29)(23,32)(25,31)(27,30)(33,37,40,36)(34,35,42,38)(39,45,48,44)(41,43,47,46);(1,5,7,11)(2,6,9,16)(3,13,14,4)(8,15,12,10)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32)(33,38,40,35)(34,37,42,36)(39,46,48,43)(41,45,47,44);(1,6)(2,11)(3,10)(4,12)(5,9)(7,16)(8,13)(14,15)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)(33,39,40,48)(34,41,42,47)(35,43,38,46)(36,44,37,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805254);