#669456

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fgi,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 576652);