#100634

order	512
nilpotency class	3
isoclinism	isoclinic class of #100555
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#29712	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

684 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ghi,d^2=gh,e^2=gi,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=egi,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7830);