#382409

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=h,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289605);