#7647613

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×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6544 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:

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

Polycyclic presentation:

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

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, 7554809);