#7631941

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	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5190 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539137);