#7899482

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

6522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7806678);