#8323596

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	#44925	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3940 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)(3,6,9,15)(4,5,8,14)(11,12,13,16)(18,20)(19,25)(24,28)(27,31)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3)(2,6)(4,11)(5,12)(7,9)(8,13)(10,15)(14,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(36,44)(39,46)(42,48)(45,47);(3,9)(6,15)(11,13)(12,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,4)(2,5)(3,11)(6,12)(7,8)(9,13)(10,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,5)(2,8)(3,12)(4,10)(6,13)(7,14)(9,16)(11,15)(17,20,22,18)(19,23,25,26)(21,28,29,24)(27,30,31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,6,7,15)(2,9,10,3)(4,12,8,16)(5,13,14,11)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8230792);