#7711957

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4656 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)(11,13,15,16)(17,18,24,30)(19,31,32,25)(20,29,23,26)(21,27,28,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,25)(20,21)(22,29)(23,28)(24,32)(26,27)(30,31);(2,5)(6,8)(17,20,24,23)(18,26,30,29)(19,28,32,21)(22,31,27,25);(17,21,24,28)(18,27,30,22)(19,20,32,23)(25,26,31,29);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,22,24,27)(18,28,30,21)(19,29,32,26)(20,25,23,31);(1,3,4,7)(2,6,5,8)(17,23,24,20)(18,29,30,26)(19,28,32,21)(22,31,27,25)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=g,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, 7619153);