#7626332

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×#3565	C₂
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)(7,8)(9,17)(10,11)(12,15)(13,18)(14,19)(16,20)(21,22,27,24)(23,26,31,28)(25,29,33,32)(30,34,36,35);(6,11)(8,12)(13,17)(16,19)(25,33)(29,32)(30,36)(34,35);(5,6,10,11)(7,12,15,8)(9,13,18,17)(14,19,20,16)(21,23,27,31)(22,28,24,26)(25,30,33,36)(29,35,32,34);(1,2,4,3)(5,7)(6,12)(8,11)(9,14)(10,15)(13,19)(16,17)(18,20)(21,24,27,22)(23,26,31,28)(25,32,33,29)(30,34,36,35);(1,3,4,2)(5,8,10,12)(6,7,11,15)(9,16,18,19)(13,14,17,20)(21,25,27,33)(22,29,24,32)(23,30,31,36)(26,34,28,35);(5,9,10,18)(6,13,11,17)(7,14,15,20)(8,16,12,19)(21,26)(22,23)(24,31)(25,34)(27,28)(29,30)(32,36)(33,35)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=gi,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, 7533528);