#8088331

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	#47760	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4329 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=ghi,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, 7995527);