#7876925

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	#38385	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 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)(9,10,14,16)(11,15,19,22)(12,20,21,17)(13,18,23,24)(25,26)(27,35)(28,34)(29,32)(30,36)(31,33)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,31,29)(26,32,33,35)(28,37,36,39)(30,38,34,40);(10,16)(11,19)(17,20)(18,24)(26,33)(28,36)(32,35)(37,39);(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28,31,36)(26,34,33,30)(27,37,29,39)(32,40,35,38);(9,12,14,21)(10,17,16,20)(11,18,19,24)(13,22,23,15)(25,29,31,27)(26,35,33,32)(28,39,36,37)(30,40,34,38);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,17)(12,15)(14,23)(16,24)(19,20)(21,22)(25,30,31,34)(26,36,33,28)(27,38,29,40)(32,39,35,37)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7784121);