#321955

order	512
nilpotency class	3
isoclinism	isoclinic class of #321935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

712 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 166 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229151);