#7716632

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4766 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)(9,10,13,12)(11,15,16,14)(17,18)(19,26)(20,22)(21,24)(23,25)(27,32)(28,31)(29,30);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,24)(20,28)(21,23)(22,29)(25,26)(27,30)(31,32);(2,5)(6,8)(18,25)(19,21)(20,27)(29,31);(9,12,13,10)(11,15,16,14)(17,20,23,27)(18,22,25,32)(19,28,21,30)(24,29,26,31);(1,2)(3,6)(4,5)(7,8)(17,21,23,19)(18,26,25,24)(20,30,27,28)(22,31,32,29);(1,3)(2,6)(4,7)(5,8)(17,22)(18,27)(19,29)(20,25)(21,31)(23,32)(24,30)(26,28)

Polycyclic presentation:

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

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, 7623828);