#231731

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	#3147	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

559 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 178 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138927);