#382533

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2114 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 2114 conjugacy classes of subgroups — too many to draw. Only its 362 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289729);