#93223

order	512
nilpotency class	5
isoclinism	isoclinic class of #93221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₄⋊C₈	C₂×C₄
Frattini, cosocle	#1985	C₂²
prime radical	#3245	C₄
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1985 ◁ G
LECS	G ▷ #1985 ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁

190 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 26 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419);