#7645619

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

5398 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,16,14,12)(19,25)(20,28)(22,26)(23,30);(9,11)(10,12)(13,14)(15,16)(17,18,21,24)(19,26,25,22)(20,23,28,30)(27,32,31,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19,21,25)(18,22,24,26)(20,27,28,31)(23,29,30,32);(2,6)(5,7)(9,12,13,16)(10,14,15,11)(17,20)(18,23)(19,27)(21,28)(22,29)(24,30)(25,31)(26,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,14)(15,16)

Polycyclic presentation:

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

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, 7552815);