#8925208

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	#47004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832404);