#8932165

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	#49740	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,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, 8839361);