#7655161

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4878 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7562357);