#238880

order	512
nilpotency class	3
isoclinism	isoclinic class of #238875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3245	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1032 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 188 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:

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

Polycyclic presentation:

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

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, 146076);