#7712621

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

6456 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)(5,6,10,7)(8,13,15,11)(9,14,17,12)(16,18,20,19)(25,33)(29,34)(30,35)(32,36);(1,2,3,4)(5,6,10,7)(8,11,15,13)(9,12,17,14)(16,18,20,19)(21,22,26,27)(23,24,28,31)(25,29,33,34)(30,32,35,36);(6,7)(8,15)(12,14)(16,20)(22,27)(24,31)(29,34)(32,36);(5,7,10,6)(8,13,15,11)(9,14,17,12)(16,19,20,18)(21,23,26,28)(22,24,27,31)(25,30,33,35)(29,32,34,36);(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,24)(22,28)(23,27)(25,32)(26,31)(29,35)(30,34)(33,36);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,25,26,33)(22,29,27,34)(23,30,28,35)(24,32,31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=i,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7619817);