#418130

order	512
nilpotency class	3
isoclinism	isoclinic class of #417496
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35390	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2505 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2505 conjugacy classes of subgroups — too many to draw. Only its 218 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,d^c=di

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, 325326);