#7651276

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7558472);