#8012012

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	#50654	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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)(9,10)(11,12)(13,21)(14,16)(15,19)(17,22)(18,23)(20,24)(25,26)(27,33)(28,32)(29,31)(30,34)(35,40)(36,39)(37,38);(1,2)(3,7)(4,8)(5,6)(9,11,14,19)(10,15,16,12)(13,18,22,24)(17,23,21,20)(25,27)(26,30)(28,36)(29,34)(31,33)(32,40)(35,38)(37,39);(1,3,5,4)(2,7,6,8)(10,16)(12,15)(17,21)(20,23)(26,31)(27,34)(32,37)(35,36);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,20)(14,15)(16,19)(17,18)(21,24)(22,23)(25,26)(27,30)(28,37)(29,31)(32,38)(33,34)(35,40)(36,39);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,23)(22,24)(25,28,29,38)(26,32,31,37)(27,35,34,36)(30,39,33,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,21)(19,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,39)(36,38)(37,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7919208);