#139982

order	512
nilpotency class	4
isoclinism	isoclinic class of #139978
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	C₂²×D₄	C₂²⋊C₄
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

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

normal subgroups only

This group has 424 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(17,18,21,23,22,27,19,25)(20,24,28,31,29,26,30,32);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15)(18,23)(19,21)(24,31)(25,27)(26,32)(28,30);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,12)(14,15)(18,24)(19,21)(20,29)(23,31)(25,32)(26,27)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=gi,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,d^b=di,e^b=ehi,f^b=fi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47178);