#140000

order	512
nilpotency class	4
isoclinism	isoclinic class of #139964
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#18166	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 568 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=egi,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, 47196);