#102404

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

700 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21,23,24)(20,22)(25,26,27,30,28,31,32,29);(2,5)(6,7)(9,11,12,10,14,16,13,15)(17,19,20,23)(18,21)(22,24)(26,29)(27,32)(30,31);(1,3,4,8)(2,6,5,7)(9,12,14,13)(10,11,15,16)(18,22)(19,23)

Polycyclic presentation:

2^9|a^2=f,b^2=defgh,c^2=eg,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,f^b=fh

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, 9600);