#93198

order	512
nilpotency class	5
isoclinism	isoclinic class of #93193
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂²×Q₈	C₂×C₈
Frattini, cosocle	#1985	C₂²
prime radical	#3497	C₄
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1985 ◁ G
LECS	G ▷ #1985 ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁

266 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,c^2=i,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ci,d^b=de,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,e^d=eh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394);