#238794

order	512
nilpotency class	3
isoclinism	isoclinic class of #234099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

976 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=dhi,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145990);