#232952

order	512
nilpotency class	3
isoclinism	isoclinic class of #232951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
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

1354 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 204 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,14)(13,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(19,26)(22,30)(27,32)(29,31)

Polycyclic presentation:

2^9|b^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,g^b=gh,f^c=fi,g^c=gh

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, 140148);