#251227

order	512
nilpotency class	3
isoclinism	isoclinic class of #249079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1066 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=hi,d^2=i,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, 158423);