#7643446

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	#35032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6634 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550642);