#7626214

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6914 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)(5,6,11,9)(7,10,14,12)(8,16,18,13)(15,19,20,17)(22,26)(24,31)(27,32)(30,35);(6,9)(7,14)(13,16)(15,20)(23,29)(27,32)(30,35)(34,36);(5,7,11,14)(6,12,9,10)(8,15,18,20)(13,19,16,17)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(5,8,11,18)(6,13,9,16)(7,15,14,20)(10,17,12,19);(5,9,11,6)(7,10,14,12)(8,16,18,13)(15,17,20,19)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(5,10)(6,7)(8,17)(9,14)(11,12)(13,15)(16,20)(18,19)(21,24)(22,28)(23,30)(25,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533410);