#254636

order	512
nilpotency class	3
isoclinism	isoclinic class of #249079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1478 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1478 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161832);