#214054

order	512
nilpotency class	3
isoclinism	isoclinic class of #214040
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37298	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

935 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 935 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121250);