#7631873

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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539069);