#7642096

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

7000 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:

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

Polycyclic presentation:

2^9|a^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549292);