#108221

order	512
nilpotency class	3
isoclinism	isoclinic class of #9471
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

757 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ch,e^b=ei,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15417);