#105406

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄×#95
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

599 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)(3,12)(4,9)(5,15)(6,10)(7,11)(8,13)(14,16)(17,18)(19,25,26,22)(20,24)(21,23)(27,31,32,29)(28,30);(1,3,5,13,7,14,6,4)(2,8,10,12,11,9,15,16)(17,19,20,27,21,26,28,32)(18,22,24,29,23,25,30,31);(1,4,6,14,7,13,5,3)(2,9,10,16,11,8,15,12)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=h,g^2=hi,b^a=bd,d^a=dh,f^a=fhi,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12602);