#7626218

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

6936 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)(6,10)(8,15)(11,16)(14,19)(21,22,26,27)(23,30,25,28)(24,29,32,34)(31,36,33,35);(7,13)(11,16)(14,19)(18,20)(22,27)(24,32)(28,30)(31,33);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,35)(32,34)(33,36);(21,23,26,25)(22,28,27,30)(24,31,32,33)(29,35,34,36);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24)(22,29)(23,31)(25,33)(26,32)(27,34)(28,35)(30,36);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25,26,23)(22,30,27,28)(24,33,32,31)(29,36,34,35)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,f^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, 7533414);