#246804

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1415 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1415 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154000);