#7674181

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581377);