#8932293

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	#53527	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=h,e^2=i,f^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, 8839489);