#379939

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1912 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)(11,14)(12,13)(15,16)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,32)(30,31);(1,2)(3,4)(10,13)(14,16)(18,22)(23,31)(24,29)(25,32);(2,4)(11,15)(14,16)(17,19)(18,23)(20,26)(21,30)(22,31)(24,32)(25,29)(27,28);(9,11,12,15)(10,14,13,16)(19,26)(23,31)(25,32)(27,30)

Polycyclic presentation:

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

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, 287135);