#150970

order	512
nilpotency class	5
isoclinism	isoclinic class of #150243
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#33733	C₂
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

224 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225429, #72225503, #72225505
maximal normal subgroup of	#72225429, #72225503, #72225505
maximal quotient of	#72225427

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58166);