#331516

order	512
nilpotency class	3
isoclinism	isoclinic class of #331095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

782 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238712);