#7626220

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	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6734 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)(6,8)(7,14)(10,18)(17,19)(22,26)(27,32)(28,33)(34,36);(5,6,11,8)(7,9,14,12)(10,13,18,16)(15,17,20,19)(24,31)(28,33)(29,35)(34,36);(5,7,11,14)(6,12,8,9)(10,15,18,20)(13,19,16,17)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(5,8,11,6)(7,9,14,12)(10,16,18,13)(15,17,20,19)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,9)(6,7)(8,14)(10,17)(11,12)(13,15)(16,20)(18,19)(21,24)(22,28)(23,29)(25,31)(26,33)(27,34)(30,35)(32,36);(5,10)(6,13)(7,15)(8,16)(9,17)(11,18)(12,19)(14,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,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, 7533416);