#8898037

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,e^2=gh,f^2=h,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, 8805233);