#106065

order	512
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1350 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1350 conjugacy classes of subgroups — too many to draw. Only its 198 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240314, #72240335, #72240337, …
maximal normal subgroup of	#72240314, #72240335, #72240337, …
maximal quotient of	#72279219, #72279289, #72286556

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13261);