#7717540

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

6682 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)(7,12)(8,10)(9,16)(11,14)(13,17)(15,18)(19,20)(25,33)(29,34)(30,35)(32,36);(1,2,3,4)(5,7)(6,11)(8,12)(9,15)(10,14)(13,18)(16,19)(17,20)(21,22)(23,24)(25,29)(26,27)(28,31)(30,32)(33,34)(35,36);(6,8)(7,14)(13,16)(15,20)(22,27)(24,31)(29,34)(32,36);(5,7)(6,12)(8,11)(9,15)(10,14)(13,19)(16,18)(17,20)(21,23,26,28)(22,24,27,31)(25,30,33,35)(29,32,34,36);(5,8,10,6)(7,11,14,12)(9,16,17,13)(15,18,20,19)(21,24)(22,28)(23,27)(25,32)(26,31)(29,35)(30,34)(33,36);(5,9)(6,13)(7,15)(8,16)(10,17)(11,18)(12,19)(14,20)(21,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,35)(31,36)

Polycyclic presentation:

2^9|b^2=gh,d^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7624736);