#191260

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

498 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 238 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=hi,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,g^b=ghi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98456);