#8057944

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965140);