#7660645

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

4714 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)(17,18,23,27)(19,20,29,24)(21,25,28,30)(22,26,31,32);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16);(1,3)(2,5)(4,8)(6,7)(17,19,23,29)(18,24,27,20)(21,22,28,31)(25,32,30,26);(1,2)(3,6)(4,7)(5,8)(17,20,23,24)(18,19,27,29)(21,26,28,32)(22,30,31,25);(1,2)(3,5)(4,7)(6,8)(9,11,12,15)(10,13,14,16)(17,21)(18,25)(19,22)(20,26)(23,28)(24,32)(27,30)(29,31);(1,3)(2,6)(4,8)(5,7)(17,22)(18,26)(19,28)(20,30)(21,29)(23,31)(24,25)(27,32)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=g,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, 7567841);