#7662438

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6591 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7569634);