#331520

order	512
nilpotency class	3
isoclinism	isoclinic class of #331095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

860 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,3,7,4,6,5,8)(10,14)(15,16)(17,18,20,23,21,19,22,24);(2,6)(7,8)(9,10,11,15,12,14,13,16)(17,19,22,23,21,18,20,24);(2,7)(3,5)(6,8)(10,15)(11,13)(14,16)(17,18)(19,21)(20,24)(22,23);(1,3,4,5)(2,7,6,8)(9,11,12,13)(10,15,14,16)(17,20,21,22)(18,23,19,24)

Polycyclic presentation:

2^9|a^2=efi,b^2=egh,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, 238716);