#449622

order	512
nilpotency class	3
isoclinism	isoclinic class of #417496
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	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2102 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 2102 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356818);