#128227

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1744
prime radical	C₂³	#431
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 242 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=g,e^2=h,h^2=i,b^a=bd,f^a=fg,c^b=ci,e^b=eg

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, 35423);