#7626478

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×#3561	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6700 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=gi,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, 7533674);