#7630603

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4668 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537799);