#7660689

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4808 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7567885);