#7642010

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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6330 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549206);