#10097153

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
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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10004349);