#128219

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1745
prime radical	C₂³	C₄×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,20,28,23,27,38,26,24,31,39,46,21,30,22,29)(19,32,35,42,34,25,40,41,37,43,48,45,36,44,33,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,33,21,36,39,48,24,37,38,40,23,34,20,35)(18,25,29,42,30,32,46,47,31,44,26,45,27,43,28,41);(3,9)(6,12)(10,15)(13,16)(17,20,21,22,24,39,23,38)(18,26,27,46,31,29,30,28)(19,33,34,35,37,40,36,48)(25,41,32,42,44,47,43,45)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=di,c^b=cg,d^b=di,e^c=ei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35415);