#7647194

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6784 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=i,e^2=i,f^2=h,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, 7554390);