#7932331

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
prime radical	#49566	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4728 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gh,f^2=gh,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, 7839527);