#7657311

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

6638 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)(5,6,11,15)(7,16,18,12)(8,10,14,19)(9,13,17,20)(21,22,27,26)(23,30,25,28)(24,32,34,29)(31,35,33,36);(1,2)(3,4)(21,23,27,25)(22,28,26,30)(24,33,34,31)(29,36,32,35);(5,7)(6,12)(8,17)(9,14)(10,13)(11,18)(15,16)(19,20)(21,24,27,34)(22,29,26,32)(23,31,25,33)(28,35,30,36);(5,8,11,14)(6,10,15,19)(7,17,18,9)(12,13,16,20)(21,25,27,23)(22,30,26,28)(24,33,34,31)(29,36,32,35);(5,9,11,17)(6,13,15,20)(7,14,18,8)(10,16,19,12)(21,26,27,22)(23,30,25,28)(24,32,34,29)(31,36,33,35);(5,10)(6,14)(7,13)(8,15)(9,12)(11,19)(16,17)(18,20)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^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, 7564507);