#7752428

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

4810 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7659624);