#128174

order	512
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#342
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₂×#342
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

557 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,e^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fh,d^b=dh,e^b=eg,h^b=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35370);