#249089

order	512
nilpotency class	3
isoclinism	isoclinic class of #249079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35150	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1466 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156285);