#7719129

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

7158 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)(7,14)(8,12)(9,11)(10,17)(13,18)(15,19)(16,20)(25,33)(28,32)(30,36)(34,35);(1,2)(3,4)(5,7,11,8)(6,12,9,14)(10,15,18,16)(13,19,17,20)(21,22)(23,27)(24,26)(25,28)(29,31)(30,34)(32,33)(35,36);(6,9)(7,8)(13,17)(15,16)(22,24)(27,29)(28,32)(34,35);(5,8,11,7)(6,12,9,14)(10,16,18,15)(13,19,17,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,9,11,6)(7,12,8,14)(10,17,18,13)(15,19,16,20)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,10)(6,13)(7,15)(8,16)(9,17)(11,18)(12,19)(14,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=i,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, 7626325);