#96849

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eh,d^2=g,e^2=i,f^2=h,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh

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, 4045);