#396444

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1646 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,f^c=fi

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, 303640);