#7709093

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	#59090	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4991 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,f^2=gh,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, 7616289);