#101180

order	512
nilpotency class	3
isoclinism	isoclinic class of #100923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

554 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,4)(7,8)(9,10,12,14,13,16,11,15)(17,18)(19,22,26,30)(20,27,28,23)(21,24)(25,29)(31,32);(1,2,4,6,5,8,3,7)(10,14)(11,12)(15,16)(17,19)(18,22,24,30)(20,25)(21,26)(23,29,27,32)(28,31);(1,3,5,4)(2,7,8,6)(17,20,21,28)(18,23,24,27)(19,25,26,31)(22,29,30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=gh,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

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, 8376);