#7853383

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	#38420	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4598 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=h,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, 7760579);