#8927045

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=g,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, 8834241);