#7766555

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

6770 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:

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

Polycyclic presentation:

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

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, 7673751);