#128173

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

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

normal subgroups only

This group has 559 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)(17,18,21,25,22,26,34,41)(19,27,29,42,30,38,43,47)(20,24,32,39,33,40,45,48)(23,35,36,31,37,44,46,28);(1,3,5,14,6,13,4,12)(2,7,10,16,9,11,8,15)(17,19)(18,23)(20,31,33,28)(21,29)(22,30)(24,38,40,27)(25,36)(26,37)(32,44,45,35)(34,43)(39,47,48,42)(41,46);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,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, 35369);