#251311

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36727	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1744 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1744 conjugacy classes of subgroups — too many to draw. Only its 150 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)(9,10)(11,13)(12,14)(15,16)(18,23)(19,26)(20,22)(24,31)(27,28)(30,32);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,24,28,30)(20,29,22,25)(26,32,27,31);(13,16)(17,19,21,28)(18,24,23,30)(20,26,22,27)(25,31,29,32);(2,6)(5,7)(17,20,21,22)(18,25,23,29)(19,27,28,26)(24,32,30,31)

Polycyclic presentation:

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

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, 158507);