#7626228

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×#3444	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6920 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)(6,10)(11,16)(12,17)(18,20)(21,22)(23,27)(24,30)(25,29)(26,28)(31,33)(32,36)(34,35);(8,15)(12,17)(13,19)(18,20)(22,23)(25,33)(28,30)(32,34);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,23,27,22)(24,30,26,28)(25,31,33,29)(32,36,34,35);(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,24,27,26)(22,28,23,30)(25,32,33,34)(29,35,31,36);(5,8)(6,12)(7,13)(9,15)(10,17)(11,18)(14,19)(16,20)(21,25,27,33)(22,29,23,31)(24,32,26,34)(28,35,30,36);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,26,27,24)(22,30,23,28)(25,34,33,32)(29,36,31,35)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=g,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, 7533424);