#7645743

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7552939);