#7655533

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

6867 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562729);