#508512

order	512
nilpotency class	4
isoclinism	isoclinic class of #508492
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

527 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,g^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=de,g^c=ghi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415708);