#7626360

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
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	C₂×#3460	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6888 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)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22)(23,25)(24,27)(26,32)(28,30)(29,34)(31,35)(33,36);(8,15)(11,13)(16,20)(18,19)(21,23)(22,28)(24,25)(26,31)(27,30)(29,35)(32,33)(34,36);(6,7)(11,13)(12,14)(18,19)(22,24)(23,30)(29,32)(31,36);(5,7,10,6)(8,13,15,11)(9,14,17,12)(16,19,20,18)(21,24,27,22)(23,25,30,28)(26,32,34,29)(31,33,36,35);(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,25)(22,23)(24,30)(26,33)(27,28)(29,31)(32,36)(34,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,26,27,34)(22,29,24,32)(23,31,30,36)(25,33,28,35)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533556);