#7658222

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

7470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565418);