#7838559

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

6842 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)(5,6,11,10)(7,12,15,20)(8,13,16,17)(9,18,19,14)(21,22,27,31)(23,28,26,30)(24,25,29,36)(32,33,34,35);(1,2)(3,4)(5,7,11,15)(6,12,10,20)(8,9,16,19)(13,18,17,14)(21,23)(22,28)(24,32)(25,33)(26,27)(29,34)(30,31)(35,36);(6,10)(7,15)(8,16)(14,18)(23,26)(28,30)(32,34)(33,35);(5,8)(6,13)(7,9)(10,17)(11,16)(12,18)(14,20)(15,19)(21,24)(22,25)(23,32)(26,34)(27,29)(28,33)(30,35)(31,36);(5,9)(6,14)(7,16)(8,15)(10,18)(11,19)(12,13)(17,20)(21,25)(22,29)(23,33)(24,31)(26,35)(27,36)(28,34)(30,32);(5,10,11,6)(7,12,15,20)(8,17,16,13)(9,18,19,14)(21,26,27,23)(22,30,31,28)(24,34,29,32)(25,35,36,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7745755);