#7626357

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

6502 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)(22,26)(24,32)(27,29)(30,36);(8,15)(11,13)(16,20)(18,19)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34);(6,7)(11,13)(12,14)(18,19)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,7,10,6)(8,13,15,11)(9,14,17,12)(16,19,20,18)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36);(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^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, 7533553);