#7626506

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×#3566	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6496 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)(5,6,9,10)(7,11,15,13)(8,16,17,12)(14,19,20,18)(21,22)(23,27)(24,25)(26,28)(29,33)(30,31)(32,34)(35,36);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23,26,32)(22,27,28,34)(24,30,29,35)(25,31,33,36);(6,10)(11,13)(12,16)(18,19)(22,28)(24,29)(27,34)(30,35);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24,26,29)(22,25,28,33)(23,30,32,35)(27,31,34,36);(1,3,4,2)(5,7,9,15)(6,11,10,13)(8,14,17,20)(12,18,16,19)(21,23)(22,27)(24,30)(25,31)(26,32)(28,34)(29,35)(33,36);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)(21,25)(22,29)(23,31)(24,28)(26,33)(27,35)(30,34)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=i,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, 7533702);