#7654763

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×#2540	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5292 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=gh,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, 7561959);