#7626329

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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×#540	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533525);