#7711928

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

4528 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)(9,10,13,12)(11,15,16,14)(17,18,22,26)(19,27,30,23)(20,24,25,21)(28,32,31,29);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,32)(29,31);(2,5)(6,8)(9,12,13,10)(11,15,16,14)(17,19,22,30)(18,23,26,27)(20,31,25,28)(21,29,24,32);(17,20,22,25)(18,24,26,21)(19,28,30,31)(23,29,27,32);(1,2)(3,6)(4,5)(7,8)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,32)(29,31);(1,3,4,7)(2,6,5,8)(17,21,22,24)(18,25,26,20)(19,29,30,32)(23,28,27,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,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, 7619124);