#150997

order	512
nilpotency class	5
isoclinism	isoclinic class of #150788
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂³	#357
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

191 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=fgh,g^2=hi,h^2=i,b^a=bd,c^a=cfi,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi

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, 58193);