#8933121

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	#48358	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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)(4,12)(7,10)(8,13)(17,18,22,23)(19,21,25,32)(20,27,29,24)(26,30,28,31)(33,34)(35,37)(36,44)(38,41)(39,43)(40,47)(42,48)(45,46);(1,2)(3,10)(4,13)(5,6)(7,9)(8,12)(11,15)(14,16)(17,19)(18,21)(20,28)(22,25)(23,32)(24,30)(26,29)(27,31)(33,35,38,43)(34,39,41,37)(36,45,47,42)(40,46,44,48);(2,6)(3,9)(8,13)(11,14)(18,23)(19,25)(20,29)(30,31)(33,35,38,43)(34,37,41,39)(36,42,47,45)(40,46,44,48);(1,3)(2,7)(4,14)(5,9)(6,10)(8,16)(11,12)(13,15)(17,20,22,29)(18,24,23,27)(19,26,25,28)(21,31,32,30)(33,35)(34,37)(36,45)(38,43)(39,41)(40,48)(42,47)(44,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,15)(9,14)(10,16)(17,21)(18,19)(20,30)(22,32)(23,25)(24,28)(26,27)(29,31)(33,36,38,47)(34,40,41,44)(35,42,43,45)(37,46,39,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,24)(22,23)(25,32)(26,31)(27,29)(28,30)(33,37)(34,35)(36,46)(38,39)(40,42)(41,43)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840317);