#174202

order	512
nilpotency class	3
isoclinism	isoclinic class of #174093
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

758 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,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, 81398);