#7626362

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	C₂²×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6674 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)(5,6,11,12)(7,8,9,13)(10,18,19,14)(15,20,17,16)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(6,12)(7,9)(14,18)(15,17)(22,26)(27,29)(28,33)(34,35);(5,7,11,9)(6,13,12,8)(10,15,19,17)(14,20,18,16)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,8,11,13)(6,7,12,9)(10,16,19,20)(14,15,18,17)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,9,11,7)(6,13,12,8)(10,17,19,15)(14,20,18,16)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35);(5,10)(6,14)(7,15)(8,16)(9,17)(11,19)(12,18)(13,20)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^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, 7533558);