#7961806

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	#37833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4871 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=hi,e^2=g,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, 7869002);