#8087975

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=h,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, 7995171);