#93104

order	512
nilpotency class	4
isoclinism	isoclinic class of #93099
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂⁵	C₄²
Frattini, prime radical, cosocle	C₂²×#121	C₂²
LCS	G ▷ C₂⁵ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂²×#121 ▷ C₂⁴ ▷ C₂² ▷ C₁

542 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,4,8)(3,6,9,13)(5,7,12,15)(10,14,11,16)(17,18)(19,25,26,22)(20,28,29,23)(21,24)(27,30)(31,32);(1,3)(2,6,7,14)(4,11,12,9)(5,10)(8,13)(15,16)(17,19,20,27,21,26,29,32)(18,22,23,30,24,25,28,31)

Polycyclic presentation:

2^9|a^2=d,b^2=e,e^2=hi,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gi,c^b=cg,d^b=df,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227676, #72227686
maximal normal subgroup of	#72227676, #72227686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300);