#7678305

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×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4776 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585501);