#8093251

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	#46318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4146 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)(3,9,11,7)(5,12,13,8)(10,14,16,15)(17,18,23,21)(19,25,20,24)(22,29,30,26)(27,31,28,32)(36,43)(40,45)(42,47)(46,48);(2,4)(7,9)(8,12)(14,15)(18,21)(19,20)(26,29)(27,28)(34,38)(35,41)(40,45)(42,47);(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000447);