#7927771

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7834967);