#7744732

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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 7651928);