#7626539

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×#3565	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6376 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)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,22)(23,31)(24,34)(25,28)(26,30)(27,29)(32,35)(33,36);(7,13)(11,17)(14,19)(18,20)(21,23,28,27)(22,29,25,31)(24,33,26,32)(30,36,34,35);(6,10)(11,17)(12,15)(18,20)(21,24,28,26)(22,30,25,34)(23,32,27,33)(29,35,31,36);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,25,28,22)(23,31,27,29)(24,30,26,34)(32,35,33,36);(1,3,4,2)(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19)(21,26,28,24)(22,30,25,34)(23,33,27,32)(29,35,31,36);(5,8,9,16)(6,12,10,15)(7,14,13,19)(11,18,17,20)(21,27,28,23)(22,31,25,29)(24,33,26,32)(30,36,34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=h,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, 7533735);