#93136

order	512
nilpotency class	4
isoclinism	isoclinic class of #93129
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#125	C₂²
prime radical	C₂⁴	#95
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×#125 ▷ C₂⁴ ▷ C₂² ▷ C₁

328 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,4,8,5,9,13,16)(3,7,10,15,11,6,12,14)(17,18)(19,27)(20,24)(21,34,35,25)(22,26)(23,28)(29,44,45,40)(30,41)(31,37)(32,47,42,38)(33,39)(36,43,48,46);(2,6,9,7)(8,15,16,14)(17,19,21,29,22,30,35,45)(18,23,25,36,26,37,34,48)(20,31,32,46,33,28,42,43)(24,27,38,40,39,41,47,44)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=h,e^2=i,b^a=bc,c^a=cf,e^a=eghi,f^a=fh,g^a=gi,c^b=cg,d^b=dfhi,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332);