#7657721

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

6640 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)(7,12)(8,16)(9,13)(10,15)(11,17)(14,18)(19,20)(21,22)(23,27)(24,31)(25,29)(26,30)(28,34)(32,36)(33,35);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(22,23)(28,31)(29,30)(35,36);(5,8,9,17)(6,11,13,16)(7,14,15,20)(10,18,12,19)(21,23,27,22)(24,31,34,28)(25,30,26,29)(32,36,33,35);(21,24,27,34)(22,28,23,31)(25,32,26,33)(29,35,30,36);(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,18)(17,20)(21,25,27,26)(22,29,23,30)(24,32,34,33)(28,35,31,36);(5,6)(7,10)(8,11)(9,13)(12,15)(14,18)(16,17)(19,20)(21,26,27,25)(22,30,23,29)(24,33,34,32)(28,36,31,35)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7564917);