#127481

order	512
nilpotency class	4
isoclinism	isoclinic class of #127217
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gh,e^2=gh,f^2=ghi,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cgi,e^b=eh,d^c=di,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308360, #72308433
maximal normal subgroup of	#72308360, #72308433

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34677);