#244093

order	512
nilpotency class	3
isoclinism	isoclinic class of #237997
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

502 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=ef,c^2=e,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151289);