#323528

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

670 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egh,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

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, 230724);