#7626512

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×#538	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6298 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)(6,8)(7,10)(13,16)(15,17)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,6,11,8)(7,14,10,12)(9,16,18,13)(15,19,17,20)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,7,11,10)(6,12,8,14)(9,15,18,17)(13,19,16,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,8,11,6)(7,12,10,14)(9,16,18,13)(15,19,17,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(1,3,4,2)(5,9)(6,13)(7,15)(8,16)(10,17)(11,18)(12,19)(14,20);(5,10,11,7)(6,12,8,14)(9,17,18,15)(13,19,16,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,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, 7533708);