#7876325

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=ghi,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, 7783521);