#120693

order	512
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

820 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gh

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, 27889);