#8077433

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	#43000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984629);