#7626345

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

6328 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=ghi,e^2=ghi,f^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, 7533541);