#139961

order	512
nilpotency class	4
isoclinism	isoclinic class of #139956
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#386
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

387 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47157);