#7743444

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

6314 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)(7,8)(9,13)(10,11)(12,16)(14,15)(17,18)(19,20)(21,22)(23,30)(24,32)(25,27)(26,28)(29,34)(31,35)(33,36);(1,2,3,4)(23,26)(28,30)(31,33)(35,36);(6,11)(7,12)(13,18)(14,19)(21,23,27,26)(22,28,25,30)(24,31,34,33)(29,35,32,36);(5,7,10,12)(6,8,11,16)(9,14,17,19)(13,15,18,20)(21,24)(22,29)(23,31)(25,32)(26,33)(27,34)(28,35)(30,36);(5,8)(6,12)(7,11)(9,15)(10,16)(13,19)(14,18)(17,20)(21,25,27,22)(23,28,26,30)(24,32,34,29)(31,35,33,36);(5,9,10,17)(6,13,11,18)(7,14,12,19)(8,15,16,20)(21,26,27,23)(22,28,25,30)(24,33,34,31)(29,35,32,36)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7650640);