#246504

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1908 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1908 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) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

(6,9)(7,11)(10,12)(14,17)(15,19)(18,20);(1,2)(3,4)(5,6,8,10)(7,12,11,9)(15,19)(17,20);(2,4)(13,14)(15,20)(16,18)(17,19);(5,6)(7,12)(8,10)(9,11)(15,19)(17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279297
maximal normal subgroup of	#72279297
maximal quotient of	#72279399

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153700);