#7729121

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

6651 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)(7,8,13,16)(9,12,17,18)(14,15,19,20)(21,22,27,29)(23,32,24,28)(25,26,31,30)(33,34,35,36);(1,2,3,4)(21,23,27,24)(22,28,29,32)(25,33,31,35)(26,36,30,34);(6,11)(7,13)(12,18)(14,19)(22,29)(23,24)(25,31)(34,36);(5,7)(6,8)(9,14)(10,13)(11,16)(12,15)(17,19)(18,20)(21,24,27,23)(22,28,29,32)(25,35,31,33)(26,36,30,34);(5,8)(6,7)(9,15)(10,16)(11,13)(12,14)(17,20)(18,19)(21,25,27,31)(22,30,29,26)(23,33,24,35)(28,34,32,36);(5,9,10,17)(6,12,11,18)(7,14,13,19)(8,15,16,20)(21,26,27,30)(22,31,29,25)(23,34,24,36)(28,33,32,35)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7636317);