#8931445

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	#35470	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8838641);