#7711949

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 7619145);