#7657304

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

6624 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)(5,6)(7,17)(8,13)(9,14)(10,15)(11,16)(12,20)(18,19)(21,22)(23,29)(24,28)(25,34)(26,30)(27,31)(32,35)(33,36);(1,2)(3,4)(21,23,28,25)(22,29,24,34)(26,32,27,33)(30,35,31,36);(5,7)(6,12)(8,18)(9,15)(10,19)(11,20)(13,14)(16,17)(21,24,28,22)(23,29,25,34)(26,31,27,30)(32,35,33,36);(5,8,11,15)(6,13,16,10)(7,18,20,9)(12,14,17,19)(21,25,28,23)(22,29,24,34)(26,33,27,32)(30,35,31,36);(5,9,11,18)(6,14,16,19)(7,15,20,8)(10,12,13,17)(21,26,28,27)(22,30,24,31)(23,32,25,33)(29,35,34,36);(5,10,11,13)(6,15,16,8)(7,19,20,14)(9,17,18,12)(21,27,28,26)(22,31,24,30)(23,33,25,32)(29,36,34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=h,f^2=h,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, 7564500);