#7642001

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	C₂×#3444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6910 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)(6,9)(7,13)(12,17)(14,19)(25,33)(29,36)(31,32)(34,35);(1,2)(3,4)(8,16)(12,17)(14,19)(18,20)(21,22,26,30)(23,28,24,27)(25,29,33,36)(31,35,32,34);(5,6,10,9)(7,11,13,15)(8,12,16,17)(14,18,19,20)(21,23,26,24)(22,27,30,28)(25,31,33,32)(29,34,36,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)(21,24,26,23)(22,28,30,27)(25,32,33,31)(29,35,36,34);(5,9,10,6)(7,15,13,11)(8,17,16,12)(14,20,19,18)(21,25)(22,29)(23,31)(24,32)(26,33)(27,34)(28,35)(30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=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, 7549197);