#7655163

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	#35064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6756 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)(5,6)(7,14)(8,16)(9,11)(10,13)(12,18)(15,19)(17,20);(1,2)(3,4)(21,22,26,29)(23,30,33,27)(24,25,28,34)(31,36,35,32);(5,7,10,9)(6,11,13,14)(8,15,18,17)(12,19,16,20)(21,23,26,33)(22,27,29,30)(24,31,28,35)(25,32,34,36);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,16)(14,20);(5,9,10,7)(6,11,13,14)(8,17,18,15)(12,19,16,20)(21,24)(22,25)(23,31)(26,28)(27,32)(29,34)(30,36)(33,35);(5,6)(7,11)(8,12)(9,14)(10,13)(15,19)(16,18)(17,20)(21,25)(22,28)(23,32)(24,29)(26,34)(27,35)(30,31)(33,36)

Polycyclic presentation:

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

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, 7562359);