#7732822

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

6739 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)(7,12)(8,16)(9,14)(10,11)(13,18)(15,17)(19,20)(21,22,26,23)(24,29,31,27)(25,28,33,30)(32,35,36,34);(1,2)(3,4)(21,22,26,23)(24,27,31,29)(25,30,33,28)(32,35,36,34);(6,11)(7,14)(13,16)(15,20)(22,23)(24,31)(28,30)(32,36);(5,7,10,14)(6,12,11,9)(8,15,18,20)(13,19,16,17)(21,23,26,22)(24,29,31,27)(25,30,33,28)(32,35,36,34);(5,8,10,18)(6,13,11,16)(7,15,14,20)(9,17,12,19)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,9,10,12)(6,14,11,7)(8,17,18,19)(13,20,16,15)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=hi,f^2=hi,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, 7640018);