#7882840

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

6658 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)(5,6)(7,8)(9,13)(10,11)(12,16)(14,15)(17,20)(18,19)(21,22,26,25)(23,27,29,31)(24,28,32,33)(30,34,35,36);(1,2,3,4)(5,6,10,11)(7,8,12,16)(9,13,18,19)(14,15,20,17)(23,29)(27,31)(30,35)(34,36);(6,11)(7,12)(13,19)(14,20)(21,23,26,29)(22,27,25,31)(24,30,32,35)(28,34,33,36);(5,7,10,12)(6,8,11,16)(9,14,18,20)(13,15,19,17);(5,8)(6,12)(7,11)(9,17)(10,16)(13,14)(15,18)(19,20)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,9,10,18)(6,13,11,19)(7,14,12,20)(8,15,16,17)(21,25,26,22)(23,31,29,27)(24,28,32,33)(30,34,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790036);